kaertner_2010_fundamentals_of_photonicskaertner 2010 fundamentals of photonics

6.602(UG)/6.621(G) Fundamentals ofPhotonics

Franz X. Kaertner

Spring Term 2009

Contents

1 Introduction 1

2 Classical Electromagnetism and Optics 132.1 Maxwell’s Equations of Isotropic Media . . . . . . . . . . . . . 13

2.1.1 Helmholtz Equation . . . . . . . . . . . . . . . . . . . 152.1.2 Plane-Wave Solutions (TEM-Waves) and Complex No-

tation . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.1.3 Poynting Vectors, Energy Density and Intensity . . . . 182.1.4 Classical Permittivity . . . . . . . . . . . . . . . . . . . 192.1.5 Optical Pulses . . . . . . . . . . . . . . . . . . . . . . . 232.1.6 Pulse Propagation . . . . . . . . . . . . . . . . . . . . 282.1.7 Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . 292.1.8 Loss and Gain . . . . . . . . . . . . . . . . . . . . . . . 372.1.9 Kramers-Kroenig Relations and Sellmeier Equation . . 39

2.2 Electromagnetic Waves and Interfaces . . . . . . . . . . . . . . 442.2.1 Boundary Conditions and Snell’s law . . . . . . . . . . 452.2.2 Measuring Refractive Index with a Prism . . . . . . . . 472.2.3 Fresnel Reflection . . . . . . . . . . . . . . . . . . . . . 502.2.4 Brewster’s Angle . . . . . . . . . . . . . . . . . . . . . 552.2.5 Total Internal Reflection . . . . . . . . . . . . . . . . . 58

2.3 Polarization of Electromagnetic Waves . . . . . . . . . . . . . 662.3.1 Polarization . . . . . . . . . . . . . . . . . . . . . . . . 662.3.2 Jones Calculus . . . . . . . . . . . . . . . . . . . . . . 71

2.4 Interference and Interferometers . . . . . . . . . . . . . . . . . 762.4.1 Interference and Coherence . . . . . . . . . . . . . . . . 762.4.2 TEM-Transmission Lines and TEM-Waves . . . . . . . 802.4.3 Scattering and Transfer Matrix . . . . . . . . . . . . . 832.4.4 Properties of the Scattering Matrix . . . . . . . . . . . 85

3

4 CONTENTS

2.4.5 Beamsplitter . . . . . . . . . . . . . . . . . . . . . . . . 862.4.6 Interferometers . . . . . . . . . . . . . . . . . . . . . . 872.4.7 Multipath Interference and Fabry-Perot Resonator . . . 912.4.8 Quality Factor of Fabry-Perot Resonances . . . . . . . 962.4.9 Thin-Film Filters . . . . . . . . . . . . . . . . . . . . . 100

2.5 Gaussian Beams and Resonators . . . . . . . . . . . . . . . . . 1022.5.1 Paraxial Wave Equation . . . . . . . . . . . . . . . . . 1022.5.2 Gaussian Beams . . . . . . . . . . . . . . . . . . . . . . 1042.5.3 Optical Resonators . . . . . . . . . . . . . . . . . . . . 111

2.6 Waveguides and Integrated Optics . . . . . . . . . . . . . . . . 1232.6.1 Planar Waveguides . . . . . . . . . . . . . . . . . . . . 1242.6.2 Two-Dimensional Waveguides . . . . . . . . . . . . . . 1432.6.3 Waveguide Coupling

1432.6.4 Coupling of Modes . . . . . . . . . . . . . . . . . . . . 1452.6.5 Switching by Control of Phase Mismatch . . . . . . . . 1502.6.6 Optical Fibers . . . . . . . . . . . . . . . . . . . . . . . 152

3 Quantum Nature of Light and Matter 161

3.1 Black Body Radiation . . . . . . . . . . . . . . . . . . . . . . 1613.1.1 Rayleigh-Jeans-Law . . . . . . . . . . . . . . . . . . . . 1653.1.2 Wien’s Law . . . . . . . . . . . . . . . . . . . . . . . . 1663.1.3 Planck’s Law . . . . . . . . . . . . . . . . . . . . . . . 1663.1.4 Thermal Photon Statistics . . . . . . . . . . . . . . . . 168

3.2 Photo-electric Effect . . . . . . . . . . . . . . . . . . . . . . . 1723.3 Photodetection and Shot Noise . . . . . . . . . . . . . . . . . 173

3.3.1 Signal to Noise Ratio . . . . . . . . . . . . . . . . . . . 1763.3.2 Noise Equivalent Power . . . . . . . . . . . . . . . . . . 177

3.4 Spontaneous and Induced Emission . . . . . . . . . . . . . . . 1773.5 Matter Waves and Bohr’s Model of an Atom . . . . . . . . . . 1813.6 Wave Particle Duality . . . . . . . . . . . . . . . . . . . . . . 186

3.7 Appendix A: Mode Counting . . . . . . . . . . . . . . . . . . . 1863.8 Appendix B: Shot Noise Formula . . . . . . . . . . . . . . . . 189

CONTENTS 5

4 Schroedinger Equation 1934.1 Free Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1964.2 Probability Conservation and Propability Currents . . . . . . 2014.3 Measureability of Physical Quantities (Observables) . . . . . . 2044.4 Stationary States . . . . . . . . . . . . . . . . . . . . . . . . . 206

4.4.1 The One-dimensional Infinite Box Potential . . . . . . 2074.4.2 The One-dimensional Harmonic Oscillator . . . . . . . 209

4.5 The Hydrogen Atom . . . . . . . . . . . . . . . . . . . . . . . 2134.5.1 Ground State . . . . . . . . . . . . . . . . . . . . . . . 2154.5.2 Excited States . . . . . . . . . . . . . . . . . . . . . . . 2184.5.3 Energy Spectrum of Hydrogen . . . . . . . . . . . . . . 2264.5.4 Superposition States, Radiative Transitions and Selec-

tion Rules . . . . . . . . . . . . . . . . . . . . . . . . . 2294.6 Wave Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . 233

4.6.1 Position Statistics . . . . . . . . . . . . . . . . . . . . . 2344.6.2 Momentum Statistics . . . . . . . . . . . . . . . . . . . 2344.6.3 Energy Statistics . . . . . . . . . . . . . . . . . . . . . 2354.6.4 Arbitrary Observable . . . . . . . . . . . . . . . . . . . 2364.6.5 Eigenfunctions and Eigenvalues of Operators . . . . . . 2374.6.6 Appendix A: Spherical Harmonics . . . . . . . . . . . . 2394.6.7 Appendix B: Radial Wave Function . . . . . . . . . . . 241

5 Interaction of Light and Matter 2475.1 The Two-Level Model . . . . . . . . . . . . . . . . . . . . . . 2475.2 The Atom-Field Interaction In Dipole Approximation . . . . . 2495.3 Rabi-Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . 2525.4 Energy- and Phase-Relaxation . . . . . . . . . . . . . . . . . . 2565.5 The Bloch Equations . . . . . . . . . . . . . . . . . . . . . . . 2615.6 Dielectric Susceptibility and Saturation . . . . . . . . . . . . . 2625.7 Rate Equations and Cross Sections . . . . . . . . . . . . . . . 265

6 Lasers 2736.1 The Laser (Oscillator) Concept . . . . . . . . . . . . . . . . . 2736.2 Laser Gain Media . . . . . . . . . . . . . . . . . . . . . . . . . 276

6.2.1 Three and Four Level Laser Media . . . . . . . . . . . 2766.3 Types of Lasers . . . . . . . . . . . . . . . . . . . . . . . . . . 279

6.3.1 Gas Lasers . . . . . . . . . . . . . . . . . . . . . . . . . 2796.3.2 Excimer Lasers: . . . . . . . . . . . . . . . . . . . . . . 280

6 CONTENTS

6.3.3 Dye Lasers: . . . . . . . . . . . . . . . . . . . . . . . . 2816.3.4 Solid-State Lasers . . . . . . . . . . . . . . . . . . . . . 2816.3.5 Semiconductor Lasers . . . . . . . . . . . . . . . . . . . 2856.3.6 Quantum Cascade Lasers . . . . . . . . . . . . . . . . . 288

6.4 Lasers and its spectroscopic parameters . . . . . . . . . . . . . 2906.5 Homogeneous and Inhomogeneous Broadening . . . . . . . . . 2916.6 Laser Dynamics (Single Mode) . . . . . . . . . . . . . . . . . . 2936.7 Continuous Wave Operation . . . . . . . . . . . . . . . . . . . 2986.8 Stability and Relaxation Oscillations . . . . . . . . . . . . . . 3006.9 Laser Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . 3036.10 "Thresholdless" Lasing

3046.11 Short pulse generation by Q-Switching . . . . . . . . . . . . . 306

6.11.1 Active Q-Switching . . . . . . . . . . . . . . . . . . . . 3076.11.2 Passive Q-Switching . . . . . . . . . . . . . . . . . . . 309

6.12 Short pulse generation by mode locking . . . . . . . . . . . . . 3106.12.1 Active Mode Locking . . . . . . . . . . . . . . . . . . . 3136.12.2 Passive Mode Locking . . . . . . . . . . . . . . . . . . 318

7 The Dirac Formalism and Hilbert Spaces 3277.1 Hilbert Space . . . . . . . . . . . . . . . . . . . . . . . . . . . 328

7.1.1 Scalar Product and Norm . . . . . . . . . . . . . . . . 3297.1.2 Vector Bases . . . . . . . . . . . . . . . . . . . . . . . . 330

7.2 Linear Operators in Hilbert Spaces . . . . . . . . . . . . . . . 3317.2.1 Properties of Linear Operators . . . . . . . . . . . . . . 3317.2.2 The Dyadic Product . . . . . . . . . . . . . . . . . . . 3337.2.3 Special Linear Operators . . . . . . . . . . . . . . . . . 3357.2.4 Inverse Operators . . . . . . . . . . . . . . . . . . . . . 3357.2.5 Adjoint or Hermitian Conjugate Operators . . . . . . . 3357.2.6 Hermitian Operators . . . . . . . . . . . . . . . . . . . 3367.2.7 Unitary Operators . . . . . . . . . . . . . . . . . . . . 3367.2.8 Projection Operators . . . . . . . . . . . . . . . . . . . 336

7.3 Eigenvalues of Operators . . . . . . . . . . . . . . . . . . . . . 3377.4 Eigenvectors of Commuting Operators . . . . . . . . . . . . . 3387.5 Complete System of Commuting Operators . . . . . . . . . . . 3397.6 Product Space . . . . . . . . . . . . . . . . . . . . . . . . . . . 3397.7 Quantum Dynamics . . . . . . . . . . . . . . . . . . . . . . . . 340

7.7.1 Schroedinger Equation . . . . . . . . . . . . . . . . . . 341

CONTENTS i

7.7.2 Schroedinger Equation in x-representation . . . . . . . 3417.7.3 Canonical Quantization . . . . . . . . . . . . . . . . . 3427.7.4 Schroedinger Picture . . . . . . . . . . . . . . . . . . . 3437.7.5 Heisenberg Picture . . . . . . . . . . . . . . . . . . . . 344

7.8 The Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . 3457.8.1 Energy Eigenstates, Creation and Annihilation Oper-

ators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3467.8.2 Matrix Representation . . . . . . . . . . . . . . . . . . 3497.8.3 Minimum Uncertainty States or Coherent States . . . . 3497.8.4 Heisenberg Picture . . . . . . . . . . . . . . . . . . . . 351

7.9 The Kopenhagen Interpretation of Quantum Mechanics . . . . 3527.9.1 Description of the State of a System . . . . . . . . . . 3527.9.2 Description of Physical Quantities . . . . . . . . . . . . 3527.9.3 The Measurement of Observables . . . . . . . . . . . . 353

8 Quantization of the Electromagnetic Field 3578.1 Decompostion of the Free Field into Normal Modes . . . . . . 3578.2 Plane Wave Expansion . . . . . . . . . . . . . . . . . . . . . . 3618.3 Quantization of the Electromagnetic Field . . . . . . . . . . . 3618.4 The 1-dimensional Case . . . . . . . . . . . . . . . . . . . . . 362

Chapter 1

Introduction

The word Photonics has been coined over the last two decades with the goalto describe the field of Optics, Optoelectronic phenomena and its applica-tions by one word similar to the very successful field of Electronics. As theword indicates, Photonics uses primarily photons to carry out its missionin contrast to Electronics which uses primarily electrons or other chargedparticles. But no strict separation of both fields is possible, at least noton an elementary level. Electrons in acceleration generate electromagneticwaves. Sometimes these waves are guided and over short distances the waveaspect can be neglected and one can talk about electronics only. Similar,pure photonics, i.e. the free electromagnetic field in vaccum, without matteris not of very much interest and use either. Quantum Mechanics teachesus that there are elementary excitations (in energy) of these waves, photonswith particle like properties. So far, we are used to thinking of electronsas classical particles but Quantum Mechanics equally assigns to them waveproperties, matter waves, like it assigns to the electromagnetic waves particlelike properties. Electrons are elementary excitations of matter waves just asphotons are elementary excitations of electromagnetic waves. We will talklater in more depth about this wave particle duality, once we have developedthe mathematical tools to analyze its meaning quantitatively.So far we have been educated using the language of classical physics.

In the classical limit, it turns out that electrons (particles with spin 1/2)are particles and photons (particles with integral spin) behave like classicalwaves and the particle nature is of vanishing importance. This is the reasonwhy Photonics eventually is experienced as more abstract than Electronics,especially if you went through three years of education primarily focused on

1

2 CHAPTER 1. INTRODUCTION

Electronics and its applications. The particle properties of electromagneticwaves become of importance when the energy of the photons considered,hf, where h is Planck’s constant and f is the frequency, is larger than thethermal energy stored in an electromagnetic mode, kT , where k is Boltz-mann’s constant and T is temperature. At room temperature this is the casefor frequencies greater than 6 THz. For lower temperatures this transitionfrequency from classical to quantum behaviour may already occur at GHzfrequencies. Certainly, at room temperature, the particle properties are im-portant in the near infrared (IR) and visible spectrum where currently thebulk of the photonic activities are carried out, see Figure 1.1.

Figure 1.1: Wavelength and frequency ranges of electromagnetic radiationand its use.

An important task of Photonics is the development of coherent sources ofradiation, which are in the optical range called LASER’s (Light Amplificationby Stimulated Emission of Radiation). The first amplifier making explicit useof the quantum properties of matter was the MASER (Microwave Amplifi-cation by Stimulated Emission of Radiation) invented by J. P. Gordon, C.H. Townes and Zeiger in 1954. The extension of the MASER principle to

3

the optical wavelength range was proposed by Schawlow and Townes in 1958and the first laser was demonstrated in 1960 by Maiman [5].As in the case of Electronics, Communications is a very important area

in Photonics, and is usually called Optical Communications, which is carriedout primarily in the infrared to visible wavelength range, see Figure 1.1. Theenormous bandwidth available at optical frequencies, roughly four orders ofmagnitude higher than typical microwave frequencies (10GHz), enables thecorresponding advance in information transmission capacity when comparedto what electronics can do over a single long distance transmission link atmicrowave frequencies. Our current society could not be sustained withoutthe worldwide deployed optical cable network, see Figure 1.2, which can beconsidered the largest connected machinery on earth. Understanding of thebasic working principles of the components used in such lightwave networks,like the optical fiber itself, waveguide couplers, modulators, light sources isat the heart of this course.

Figure 1.2: Worldwide underwater optical cable systems

The big advantage of Photonics versus Electronics in the area of infor-mation transmission is that photons do not interact with each other directly.Maxwell’s equations in the classical vacuum are linear. Whereas electrons,


which are charged particles, show very strong interaction via their Coulombfields. This is one reason why Photonics is very powerful in information trans-mission; essentially no interference with other signals; whereas electronics isvery strong in information processing which is based on nonlinear operations.

Figure 1.3: High-Index Contrast SiN/SiO2/Air Filter using Scanning Elec-tron Beam Lithograph. Courtesy of H. Smith MIT-Nanostructures Labora-tory

As in the case of Electronics, miniaturization of many components andintegration on a common technology platform is one of the major challengesin photonics (Integrated Optics). Over the last few years a new field calledSilicon-Photonics came to life. Modern nanofabrication techniques such asScanning Electron Beam Lithography enable the fabrication of optical com-ponents on the scale of the optical wavelength with a relative precision inthe picometer range in the same material system like electronics, i.e. sili-con/silicon nitride/silicon dioxide, see Figure 1.3. Fig. 1.4 shows a measured

5

filter characteristic useful for optical communications applications. Systemsthat integrate both optical and electronic subsystems on a single chip calledElectronic-Photonic Integrated Circuits are on the horizon.

in through

adddrop

stage through

discard

in through

adddrop

stage through

discard

Figure 1.4: Schematic and measured two stage third order ring filter. Cour-tesy MIT-Optics and Quantum Electronics Group

As with Electronics, Photonics has not only left its mark on communi-cations but is used in a wide variety of sensing applications specific to itswavelength range as well as optical imaging, test and measurement instru-mentation, manufacturing and materials processing. High power lasers withaverage powers of tens of kilowatts can cut many centimeter thick steel platesat high speed. Laser systems reaching petawatts (1015Watt) of peak powerhave been built at Lawrence Livermore Laboratory for the first time in 1996,(see Fig. 1.5) and much more compact versions are under construction atseveral laboratories around the world. The enormous peak power (the av-erage power consumption of the earth is on the order of a few Terawatt(1012Watt)), even though only available over a fraction of a picosecond, willenable us to investigate new physics and fundamental interactions at extremeintensities, such as the scattering of photons with each other invoking vacuumnonlinearities (the Quantum Electrodynamic Vacuum), see Figure 1.6.


Figure 1.5: First petawatt laser system installed at LLNL us-ing chirped pulse amplification and the NOVA amplifier chain, seehttp://www.llnl.gov/str/MPerry.html.

Because of the higher operating frequencies, optical sources can providemuch higher time resolution than microwave sources. Therefore, it is notsurprising that the shortest controlled events that currently can be made areoptical pulses that comprise only a few cycles of light. These events are soshort that one can only use the pulses themselves to reveal its pulse width,for example by performing an autocorrelation of the pulse with a copy ofitself using a nonlinear element, see Fig. 1.7.

7

Figure 1.6: Progress in peak intensity generation from lasers over the last 45years. Courtesy Gerard Mourou.


Figure 1.7: Experimental setup for measureing an interferometric autocorre-lation. A copy of the pulse is generated and interferometrically delayed withrespect to the original pulse. The superposition of the two pulses undergosecond harmonic generation in a crystal and the second harmonic light isdetected.

Figure 1.8 shows a measured interferometric second harmonic autocorrela-tion trace of a 5fs (fs=femtosecond, 10−15s) short pulse at a center frequencyof 800nm [19].

8

6

4

2

0

IAC

3020100-10-20TIME DELAY, FS

Figure 1.8: Measured and fitted interferometric autocorrelation function ofa 5fs pulse [19].

Femtosecond pulses have been used to understand the carrier dynamicsin semiconductors, that limit the speed of electronic devices, the dynamics

9

of chemical reactions such as photosynthesis or the early stages of vision.As in any discipline the development does not stop and most recently pulsesas short as 250 attoseconds (10−18s) have been generated in the soft-x-rayregime [4]. Laser like sources potentially generating radiation up into thehard x-ray regime are being planned to be built over the next years [5]. Thehope is that such sources enable the spatially and temporally resolved directimaging of molecules and atoms undergoing reactions, which would advancemedical diagnostics and drug discovery. As these examples show, Photonicsis a booming field. The basis for its success is fundamental and, therefore, itwill continue to grow in importance with impact in our everyday lifes as hasbeen the case with (Micro-) Electronics, just think about your high-speedinternet connection or the CD-player.The course is organized as follows: In Chapter 2 we will review some con-

cepts of Classical Electromagnetism from 6.013, such as plane electromag-netic waves in isotropic media, the related energy flow, and their propertiesupon reflections at interfaces. This will enable us to study various importantoptical devices such as prisms, interferometers, Fabry-Perot resonators andthin film interference coatings. Optics in crystalline materials and relatedphenomena such as birefringence and its multiple use in optical componentsfor polarization manipulation will be discussed. The plane electromagneticwave is an idealization similar to the perfect monochromatic sin or cosine insignal theory. We will use the plane waves to understand beams of finite sizeand discuss them as solutions to the paraxial wave equation, the Gaussianbeams which in the limit of large beam size become optical rays. Then wewill look into guided optical beams such as in optical fibers and waveguidesfrom the point of view of ray optics and from the view point of full solutionsto Maxwell’s equations. The concept of optical modes and resonances willbe discussed.This basic training in wave physics puts us in an excellent position to

understand Quantum Mechanics, which introduces matter waves. We willrevisit in chapter III the expermimental facts appearing at the turn of the19th century that lead to the discovery of Quantum Mechanics. In chapterIV, we will find the Schroedinger Equation, a wave equation for matter waves,and introduce the formalism of Quantum Mechanics at the example of theharmonic oscillator and the hydrogen atom. Both systems are model systemsfor understanding photons and energy eigenspectra of quantum systems thatcan be fully solved analytically.Chapter V is devoted to the basic understanding of the interaction of light


and matter and sets the basis for the study of the laser and its fundamentalproperties in chapter VI. In chapter VII the modulation of light using electroand acousto-optic effects is discussed. If time permits an introduction to thequantum theory of light fields and photo detection is also treated.

Bibliography

[1] T. H. Maimann, "Stimulated optical radiation in ruby", Nature 187,493-494, (1960).

[2] M. D. Perry et al., "Petawatt Laser Pulses," Optics Letters, 24, 160-163(1999).

[3] R. Ell, U. Morgner, F.X. Kärtner, J.G. Fujimoto, E.P. Ippen, V. Scheuer,G. Angelow, T. Tschudi: Generation of 5-fs pulses and octave-spanningspectra directly from a Ti:Sappire laser, Opt. Lett. 26, 373-375 (2001)

[4] H. Hentschel, R. Kienberger, Ch. Spielmann, G. A. Reider, N. Milosevic,T. Brabec, P. Corkum, U. Heinzmann, M. Drescher, F. Krausz: "Attosec-ond Metrology," Nature 414, 509-513 (2001).

[5] John Arthur, "Status of the LCLS x-ray FEL program," Review of Sci-entific Instruments, 73, 1393-1395 (2002).

11

12 BIBLIOGRAPHY

Chapter 2

Classical Electromagnetism andOptics

The classical electromagnetic phenomena are completely described byMaxwell’sEquations. The simplest case we may consider is that of electrodynamics ofisotropic media

2.1 Maxwell’s Equations of Isotropic Media

Maxwell’s Equations are

∇×H =∂D

∂t+ J, (2.1a)

∇×E = −∂B∂t

, (2.1b)

∇ ·D = ρ, (2.1c)

∇ ·B = 0. (2.1d)

The material equations accompanying Maxwell’s equations are:

D = 0E + P, (2.2a)

B = μ0H +M. (2.2b)

Here, E and H are the electric and magnetic field, D the dielectric flux, Bthe magnetic flux, J the current density of free chareges, ρ is the free chargedensity, P is the polarization, and M the magnetization.

13

14 CHAPTER 2. CLASSICAL ELECTROMAGNETISM AND OPTICS

Note, it is Eqs.(2.2a) and (2.2b) which make electromagnetism an inter-esting and always a hot topic with never ending possibilities. All advancesin engineering of artifical materials or finding of new material properties,such as superconductivity and meta-materials, bring new life, meaning andpossibilities into this field.By taking the curl of Eq. (2.1b) and considering

∇×³∇×E

´= ∇

³∇ · E

´−∆E,

where ∇ is the Nabla operator and ∆ the Laplace operator, we obtain

∆E − μ0∂

∂t

Ãj + 0

∂E

∂t+

∂P

∂t

!=

∂

∂t∇×M+∇

³∇ ·E

´(2.3)

and henceµ∆− 1

c20

∂2

∂t2

¶E = μ0

Ã∂j

∂t+

∂2

∂t2P

!+

∂

∂t∇×M+∇

³∇ · E

´. (2.4)

with the vacuum velocity of light

c0 =

s1

μ0 0. (2.5)

For dielectric non magnetic media, which we often encounter in optics,with no free charges and currents due to free charges, there isM = 0, J = 0,ρ = 0. One can also show that the electric field can be decomposed into alongitudinal and tranasversal component EL and ET , which are characterizedby [6]

∇×EL = 0 and ∇ · ET = 0 (2.6)

If there are no free charges, the longitudinal component is zero and only atransversal component is left over. Therefore, for the purpose of this class(and most of optics) the wave equation greatly simplifies toµ

∆− 1

c20

∂2

∂t2

¶E = μ0

∂2

∂t2P. (2.7)

This is the wave equation driven by the polarization of the medium in whichthe field propagates.

2.1. MAXWELL’S EQUATIONS OF ISOTROPIC MEDIA 15

2.1.1 Helmholtz Equation

In general, the polarization in dielectric media may have a nonlinear andnon local dependence on the field. For linear media the polarizability of themedium is described by a dielectric susceptibility χ (r, t)

P (r, t) = 0

Z Zdr0dt0 χ (r − r0, t− t0)E (r0, t0) . (2.8)

The polarization in media with a local dielectric suszeptibility can be de-scribed by

P (r, t) = 0

Zdt0 χ (r, t− t0)E (r, t0) . (2.9)

This relationship further simplifies for homogeneous media, where the sus-ceptibility does not depend on location

P (r, t) = 0

Zdt0 χ (t− t0)E (r, t0) . (2.10)

which leads to a dielectric response function or permittivity

(t) = 0(δ(t) + χ (t)) (2.11)

and with it toD(r, t) =

Zdt0 (t− t0)E (r, t0) . (2.12)

If the medium is linear and has only an induced polarization, completelydescribed in the time domain χ (t) or in the frequency domain by its Fouriertransform, the complex susceptibility χ(ω) = r(ω) − 1 with the relativepermittivity r(ω) = ˜(ω)/ 0, we obtain in the frequency domain with theFourier transform relationship

eE(r, ω) =

+∞Z−∞

E(r, t)e−jωtdt, (2.13)

eP (r, ω) = 0χ(ω)

eE(r, ω), (2.14)

where, the tildes denote the Fourier transforms in the following. Substitutedinto (2.7) µ

∆+ω2

c20

¶ eE(r, ω) = −ω2μ0 0χ(ω)

eE(r, ω), (2.15)


we obtain µ∆+

ω2

c20(1 + χ(ω)

¶ eE(r, ω) = 0, (2.16)

with the refractive index n(ω) and 1+ χ(ω) = n(ω)2 results in the Helmholtzequation µ

∆+ω2

c2

¶ eE(r, ω) = 0, (2.17)

where c(ω) = c0/n(ω) is the velocity of light in the medium. This equationis the starting point for finding monochromatic wave solutions to Maxwell’sequations in linear media, as we will study for different cases in the following.So far we have treated the susceptibility χ(ω) as a real quantity, which maynot always be the case as we will see later in detail.

2.1.2 Plane-Wave Solutions (TEM-Waves) and Com-plex Notation

The real wave equation (2.7) for a linear medium has real monochromaticplane wave solutions Ek(r, t), which can be be written most efficiently interms of the complex plane-wave solutions Ek(r, t) according to

Ek(r, t) =1

2

hEk(r, t) +Ek(r, t)

∗i= <e

nEk(r, t)

o, (2.18)

withEk(r, t) = Ek e

j(ωt−k·r) e(k). (2.19)

Note, we explicitly underlined the complex wave to indicate that this is acomplex quantity. Here, e(k) is a unit vector indicating the direction of theelectric field which is also called the polarization of the wave, and Ek isthe complex field amplitude of the wave with wave vector k. Substitutionof eq.(2.18) into the wave equation results in the dispersion relation, i.e. arelationship between wave vector and frequency necessary to satisfy the waveequation

|k|2 = ω2

c(ω)2= k(ω)2. (2.20)

The relation between wave vector and frequency of a wave is called dispersionrelation. Here, it is given by

k(ω) = ±ω

c0n(ω). (2.21)


with the wavenumberk = 2π/λ, (2.22)

where λ is the wavelength of the wave in the medium with refractive indexn, ω the angular frequency, k the wave vector. Note, the natural frequencyf = ω/2π. From ∇ · E = 0, for all time, we see that k ⊥ e. Substitution ofthe electric field 2.18 into Maxwell’s Eqs. (2.1b) results in the magnetic field

Hk(r, t) =1

2

hHk(r, t) +Hk(r, t)

∗i

(2.23)

withHk(r, t) = Hk e

j(ωt−k·r) h(k). (2.24)

This complex component of the magnetic field can be determined from thecorresponding complex electric field component using Faraday’s law

−jk ×³Ek e

j(ωt−k·r) e(k)´= −jμ0ωHk(r, t), (2.25)

or

Hk(r, t) =Ek

μ0ωej(ωt−k·r)k × e = Hke

j(ωt−k·r)h (2.26)

with

h(k) =k

|k| × e(k) (2.27)

and

Hk =|k|μ0ω

Ek =1

ZFEk. (2.28)

The characteristic impedance of the TEM-wave is the ratio between electricand magnetic field strength

ZF = μ0c =

rμ0

0 r=1

nZF0 (2.29)

with the refractive index n =√

r and the free space impedance

ZF0 =

rμ0

0≈ 377Ω. (2.30)

Note that the vectors e, h and k form an orthogonal trihedral,

e ⊥ h, k ⊥ e, k ⊥ h. (2.31)


That is why we call these waves transverse electromagnetic (TEM) waves.We consider the electric field of a monochromatic electromagnetic wave withfrequency ω and electric field amplitude E0, which propagates in vacuumalong the z-axis, and is polarized along the x-axis, (Fig. 2.1), i.e. k

|k| = ez,

and e(k) = ex. Then we obtain from Eqs.(2.18) and (2.19)

E(r, t) = E0 cos(ωt− kz) ex, (2.32)

and similiar for the magnetic field

H(r, t) =E0ZF0

cos(ωt− kz) ey, (2.33)

see Figure 2.1.

c

y

x

z

E

H

Figure 2.1: Transverse electromagnetic wave (TEM) [6]

Note, that for a backward propagating wave with E(r, t) = E ejωt+jk·r ex,

and H(r, t) = H ej(ωt+kr) ey, there is a sign change for the magnetic field

H = − |k|μ0ω

E, (2.34)

so that the (k,E,H) always form a right handed orthogonal system.

2.1.3 Poynting Vectors, Energy Density and Intensity

The table below summarizes the instantaneous and time averaged energycontent and energy transport related to an electromagnetic field


Quantity Real fields Complex fields

Electric andmagnetic energydensity

we =12E ·D = 1

2 0 rE2

wm =12H ·B = 1

2μ0μrH

2

w = we + wm

hwei = 14 0 r

¯E¯2

hwmi = 14μ0μr

¯H¯2

hwi = hwei+ hwmiPoynting vector S = E×H T= 1

2E×H∗

Poynting theorem divS +E · j + ∂w∂t= 0

divT + 12E · j∗+

+2jω(hwmi− hwei) = 0Intensity I =

¯S¯= cw I = ReT = c hwi

Table 2.1: Poynting vector and energy density in EM-fields

For a plane wave with an electric field E(r, t) = Eej(ωt−kz) ex in a losslessmedium, i.e. r = real,we obtain for the time averaged energy density inunits of [J/m3]

hwi = 1

2r 0|E|2, (2.35)

the complex Poynting vector

T =1

2ZF|E|2 ez, (2.36)

and the intensity in units of [W/m2]

I =1

2ZF|E|2 = 1

2ZF |H|2. (2.37)

2.1.4 Classical Permittivity

In this section we want to get insight into propagation of an electromagneticwavepacket in an isotropic and homogeneous medium, such as a glass opticalfiber due to the interaction of radiation with the medium. The electromag-netic properties of a dielectric medium is largely determined by the electricpolarization P (t) induced by an electric field in the medium. The polariza-tion is defined as the total induced dipole moment per unit volume. If p isthe dipole moment of the elementary unit (atom, molecule, ...) constitutingthe medium and N is density of elementary units, then the polirization is

P (t) =dipole moment

volume= N · p(t), (2.38)


In the frequency domain, i.e. after Fourier transformation we obtain

eP (ω) =

dipole momentvolume

= N · ep(ω) = 0eχ(ω)eE(ω), (2.39)

assuming again a linear plarizability proportional to the electric field. Orexpressed in a different way the dielectric suszeptibility is the frequency re-sponse function of the polarization in a medium to an applied field

eχ(ω) = N · ep(ω)0eE(ω)

. (2.40)

Atoms and Molecules are composed of charge particles, such as a possitivelycharged nucleus and electrons bound to the nucleus by the Coulomb force.As a first approximation towards the interaction of light and matter we con-sider a simple model for matter, where the elementary units, the atoms ormolecules are modelled by a positively charge nucleus and an electron boundto it by a force increasing linear with distance between nucleus and electron,see Figure 2.2.

Figure 2.2: Classical harmonic oscillator model for radiation matter interac-tion

As it turns out (justification later) this simple model correctly describesmany aspects of the interaction of light with matter at very low electric fieldstrength, i.e. the fields do not change the electron distribution in the atomconsiderably or even ionize the atom. This model is called Lorentz modelafter the famous physicist A. H. Lorentz (Dutchman) studying electromag-netic phenomena at the turn of the 19th century. He also found the LorentzTransformation and Invariance of Maxwell’s Equations with respect to thesetransformation, which showed the path to Special Relativity.


The equation of motion for such a unit is the damped harmonic oscillatordriven by an electric field in one dimension, x. At optical frequencies, thedistance of elongation, x, is much smaller than an optical wavelength (atomshave dimensions on the order of a tenth of a nanometer, whereas optical fieldshave wavelength on the order of microns) and, therefore, we can neglectthe spatial variation of the electric field during the motion of the chargeswithin an atom (dipole approximation, i.e. E(r, t) = E(rA, t) = E(t)ex. Theequation of motion is then.

md2x

dt2+ 2

Ω0Qmdx

dt+mΩ20x = e0E(t), (2.41)

where E(t) = Eejωt. Here, m is the mass of the electron assuming the thatthe rest atom has infinite mass, e0 the charge of the electron, Ω0 is theresonance frequency of the undamped oscillator and Q the quality factor ofthe resonance, which determines the damping of the oscillator. By using thetrial solution x(t) = xejωt, we obtain for the complex amplitude of the dipolemoment p with the time dependent response p(t) = e0x(t) = pejωt

p =e20m

(Ω20 − ω2) + 2jΩ0QωE. (2.42)

Note, that we included ad hoc a damping term in the harmonic oscillatorequation. At this point it is not clear what the physical origin of this dampingterm is and we will discuss this at length later in chapter 4. For the moment,we can view this term simply as a consequence of irreversible interactions ofthe atom with its environment. The simplest damping mechanism is that anaccelerated electron radiates, i.e. radiation damping. We then obtain from(2.39) for the susceptibility

eχ(ω) = Ne20m10

(Ω20 − ω2) + 2jωΩ0Q

(2.43)

or eχ(ω) = ω2p

(Ω20 − ω2) + 2jωΩ0Q

, (2.44)

where ωp is called the plasma frequency defined as ω2p = Ne20/m 0. Themeaning of the plasma frequency will be further elucidated in recitations


and on problem sets. Figure 2.3 shows the real and imaginary part of theresulting classical dielectric susceptibility, eχ(ω) = eχr(ω) + jeχi(ω), (2.44),normalized to the absolute value of the imaginary part at ω = Ω0, which isω2pQ/ (2Ω

20).

Figure 2.3: Real part (dashed line) and imaginary part (solid line) of thesusceptibility of the classical oscillator model for the dielectric polarizability.

Note, that there is a small resonance shift (almost invisible) due to theloss. Off resonance, the imaginary part approaches zero very quickly. Notso the real part, which approaches a constant value ω2p/Ω

20 below resonance

for ω → 0, and approaches zero far above resonance, but much slower thanthe imaginary part. As we will see later, this is the reason why there are lowloss, i.e. transparent, media with refractive index very much different from1, because as discussed above the real part of the dielectric susceptibilitycontributes to the refractive index in a material. The negative imaginarypart describes obsorption of the electromagnetic wave in the medium madeup of damped harmonic oscillators driven by the electric field of the electro-magnetic wave. Strong absorption occurs when the driving frequency is onresonance with the damped harmonic oscillator, i.e. the absorption resonanceof the medium.After having a model for the dielectric susceptibility of a medium, we can


study how optical signals may propagate in such a medium, most importantlyoptical pulses, such as those used in optical communications.

2.1.5 Optical Pulses

Optical pulses are wave packets constructed by a continuous superpositionof monochromatic plane waves. Consider a TEM-wavepacket, i.e. a super-position of waves with different frequencies, polarized along the x-axis andpropagating along the z-axis

E(r, t) =

Z ∞

0

dΩ

2πeE(Ω)ej(Ωt−K(Ω)z) ex. (2.45)

Correspondingly, the magnetic field is given by

H(r, t) =

Z ∞

0

dΩ

2πZF (Ω)eE(Ω)ej(Ωt−K(Ω)z) ey (2.46)

Again, the physical electric and magnetic fields are real and related to thecomplex fields by

E(r, t) =1

2

³E(r, t) +E(r, t)∗

´(2.47)

H(r, t) =1

2

³H(r, t) +H(r, t)∗

´. (2.48)

Here, |E(Ω)|ejϕ(Ω) is the complex wave amplitude of the electromagnetic waveat frequency Ω and K(Ω) = Ω/c(Ω) = n(Ω)Ω/c0 the wavenumber, where,n(Ω) is again the refractive index of the medium

n2(Ω) = 1 + χ(Ω), (2.49)

c and c0 are the velocity of light in the medium and in vacuum, respectively.The planes of constant phase propagate with the phase velocity c(Ω) of thewave depending on frequency in a dispersive medium.The wavepacket consists of a superposition of many frequencies with the

spectrum shown in Fig. 2.4.At a given point in space, for simplicity z = 0, the complex field of a

pulse is given by (Fig. 2.4)

E(z = 0, t) =1

2π

Z ∞

0

E(Ω)ejΩtdΩ. (2.50)


ω0

Ω

|E( )|Ω

|A( )|ω

ω0

Figure 2.4: Spectrum of an optical wave packet described in absolute andrelative frequencies

Optical pulses often have relatively small spectral width compared to thecenter frequency of the pulse ω0, as it is illustrated in the upper part of Figure2.4. For example typical pulses used in optical communication systems for10Gb/s transmission speed are 20 ps in lenght and have a center wavelengthof λ = 1550 nm. Thus the spectral width is only on the order of 50 GHz,whereas the center frequency of the pulse is 200 THz, i.e. the bandwidth is4000 smaller than the center frequency. In such cases it is useful to separatethe complex electric field in Eq. (2.50) into a carrier frequency ω0 and anenvelope A(t) and represent the absolute frequency as Ω = ω0 + ω. We canthen rewrite Eq.(2.50) as

E(z = 0, t) =1

2π

Z ∞

−ω0E(ω0 + ω)ej(ω0+ω)tdω (2.51)

=1

2πejω0t

Z ∞

−ω0E(ω0 + ω)ejωtdω

A(t)ejω0t. (2.52)


The envelope, see Figure 2.8, is given by

A(t) =1

2π

Z ∞

−ω0→−∞A(ω)ejωtdω (2.53)

=1

2π

Z ∞

−∞A(ω)ejωtdω, (2.54)

where A(ω) = E(ω0+ω) is the spectrum of the envelope with, A(ω) = 0 forω ≤ −ω0. To be physically meaningful, the spectral amplitude A(ω) must bezero for negative frequencies less than or equal to the carrier frequency, seeFigure 2.8. Note, that waves with zero frequency can not propagate, sincethe corresponding wave vector is zero. The pulse and its envelope are shownin Figure 2.5.

Figure 2.5: Electric field and envelope of an optical pulse.

Table 2.2 shows pulse shape and spectra of some often used pulses as wellas the pulse width and time bandwidth products. The pulse width and band-width are usually specified as the Full Width at Half Maximum (FWHM) of

the intensity in the time domain, |A(t)|2 , and the spectral density¯A(ω)

¯2in

the frequency domain, respectively. Pulse shapes and corresponding spectrato the pulses listed in Table 2.2 are shown in Figs 2.6 and 2.7.


Pulse Shape Fourier TransformPulseWidth

Time-Band-width Product

A(t) A(ω) =R∞−∞ a(t)e−jωtdt ∆t ∆t ·∆f

Gaussian: e−t2

2τ2√2πτe−

12τ2ω2 2

√ln 2τ 0.441

Hyperbolic Secant:sech( t

τ)

τ2sech

¡π2τω¢

1.7627 τ 0.315

Rect-function:½1, |t| ≤ τ/20, |t| > τ/2

τ sin(τω/2)τω/2

τ 0.886

Lorentzian: 11+(t/τ)2

2πτe−|τω| 1.287 τ 0.142

Double-Exp.: e−| tτ | τ1+(ωτ)2

ln2 τ 0.142

Table 2.2: Pulse shapes, corresponding spectra and time bandwidth prod-ucts.

Figure 2.6: Fourier transforms to pulse shapes listed in table 2.2 [6].


Figure 2.7: Fourier transforms to pulse shapes listed in table 2.2 continued[6].


2.1.6 Pulse Propagation

Having a basic model for the interaction of light and matter at hand, viasection 2.1.4, we can investigate what happens if an electromagnetic wavepacket, i.e. an optical pulse propagates through such a medium. We startfrom Eqs.(2.45) to evaluate the wave packet propagation for an arbitrarypropagation distance z

E(z, t) =1

2π

Z ∞

0

E(Ω)ej(Ωt−K(Ω)z)dΩ. (2.55)

Analogous to Eq. (2.51) for a pulse at a given position, we can separate anoptical pulse into a carrier wave at frequency ω0 and a complex envelopeA(z, t),

E(z, t) = A(z, t)ej(ω0t−K(ω0)z). (2.56)

By introducing the offset frequency ω, the offset wavenumber k(ω) and spec-trum of the envelope A(ω)

ω = Ω− ω0, (2.57)

k(ω) = K(ω0 + ω)−K(ω0), (2.58)

A(ω) = E(Ω = ω0 + ω), (2.59)

we can rewrite Eq.(2.55) as

E(z, t) =1

2π

Z ∞

−∞A(ω)ej(ωt−k(ω)z)dω ej(ω0t−K(ω0)z). (2.60)

Thus the envelope at propagation distance z, see Fig.2.8, is expressed as

A(z, t) =1

2π

Z ∞

−∞A(ω)ej(ωt−k(ω)z)dω, (2.61)

with the same constraints on the spectrum of the envelope as before, i.e. thespectrum of the envelope must be zero for negative frequencies beyond thecarrier frequency. In the frequency domain Eq.(2.61)) corresponds to

A(z, ω) = A(z = 0, ω)e−jk(ω)z. (2.62)

Depending on the dispersion relation k(ω), (see Fig. 2.9),.the pulse willbe reshaped during propagation as discussed in the following section.


Figure 2.8: Electric field and pulse envelope in time domain.

Figure 2.9: Taylor expansion of dispersion relation at the center frequencyof the wave packet.

2.1.7 Dispersion

The dispersion relation K(Ω) or differential propagation constant, k(ω), in-dicates how much phase shift each frequency component experiences during


propagation. These phase shifts, if not linear with respect to frequency, willlead to distortions of the pulse. If the dispersion relation K(Ω) is only slowlyvarying over the pulse spectrum, it is useful to represent it or by its Taylorexpansion, see Fig. 2.9,

k(ω) = k0ω +k00

2ω2 +

k(3)

6ω3 +O(ω4). (2.63)

If the refractive index depends on frequency, the dispersion relation is nolonger linear with respect to frequency, see Fig. 2.9 and the pulse propagationaccording to (2.61) can be understood most easily in the frequency domain

∂A(z, ω)

∂z= −jk(ω)A(z, ω). (2.64)

Transformation of Eq.(2.64) into the time domain gives

∂A(z, t)

∂z= −j

∞Xn=1

k(n)

n!

µ−j ∂

∂t

¶n

A(z, t). (2.65)

If we keep only the first term, the linear term, in Eq.(2.62), then we obtainfor the pulse envelope from (2.61)

A(z, ω) = A(z = 0, ω)e−jk0ωz. (2.66)

According to the shift theorem for Fourier Transfroms, a linear phase overthe spectrum corresponds to a shift in the time domain,

A(z, t) = A(0, t− z/υg0), (2.67)

where we introduced the group velocity at frequency ω0

υg0 = 1/k0 =

µdk(ω)

dω

¯ω=0

¶−1=

ÃdK(Ω)

dΩ

¯Ω=ω0

!−1(2.68)

Thus the derivative of the dispersion relation at the carrier frequency deter-mines the propagation velocity of the envelope of the wave packet or groupvelocity, whereas the ratio between propagation constant and frequency de-termines the phase velocity of the carrier

υp0 = ω0/K(ω0) =

µK(ω0)

ω0

¶−1. (2.69)


To get rid of the trivial motion of the pulse envelope with the group velocity,we introduce the retarded time t0 = t− z/vg0. With respect to this retardedtime the pulse shape is invariant during propagation, if we approximate thedispersion relation by the slope at the carrier frequency

A(z, t) = A(0, t0). (2.70)

Note, if we approximate the dispersion relation by its slope at the carrierfrequency, i.e. we retain only the first term in Eq.(2.65), we obtain

∂A(z, t)

∂z+1

υg0

∂A(z, t)

∂t= 0, (2.71)

and (??) is its solution. If, we transform this equation to the new coordinatesystem

z0 = z, (2.72)

t0 = t− z/υg0, (2.73)

with

∂

∂z=

∂

∂z0− 1

υg0

∂

∂t0, (2.74)

∂

∂t=

∂

∂t0(2.75)

the transformed equation is

∂A(z0, t0)

∂z0= 0. (2.76)

Thus we see that the pulse shape doesn’t change during propagation. Thepropagation can be simply accounted for by introducting a retarded timet0 = t− z/υg0. Since z is equal to z0 we keep z in the following.If the spectrum of the pulse is broad enough or the propagation distance

long enough, so that the second order term in (2.63) becomes important, thepulse will no longer keep its shape. When keeping in the dispersion relationterms up to second order it follows from (2.65) and (2.72,2.73)

∂A(z, t0)

∂z= j

k00

2

∂2A(z, t0)

∂t02. (2.77)


This is the first non trivial term in the wave equation for the envelope.Because of the superposition principle, the pulse can be thought of to bedecomposed into wavepackets (sub-pulses) with different center frequencies.Now, the group velocity depends on the spectral component of the pulse, seeFigure 2.10, which will lead to broadening or dispersion of the pulse.

A( )ω

Dis

pers

ion

Rel

atio

n

ω0 ω1−ω1

k”ω212

Spec

trum

1 2 3

∼Δvg1 ∼Δvg3

∼Δvg2

Figure 2.10: Decomposition of a pulse into wave packets with different centerfrequency. In a medium with dispersion the wavepackets move at differentrelative group velocity.

Fortunately, for a Gaussian pulse, the pulse propagation equation 2.77can be solved analytically. The initial pulse is then of the form

E(z = 0, t) = A(z = 0, t)ejω0t (2.78)

A(z = 0, t = t0) = A0 exp

∙−12

t02

τ 2

¸(2.79)

Eq.(2.77) is most easily solved in the frequency domain since it transformsto

∂A(z, ω)

∂z= −jk

00ω2

2A(z, ω), (2.80)

with the solution

A(z, ω) = A(z = 0, ω) exp

∙−jk

00ω2

2z

¸. (2.81)


The pulse spectrum aquires a parabolic phase. Note, that here ω is theFourier Transform variable conjugate to t0 rather than t. The Gaussian pulsehas the advantage that its Fourier transform is also a Gaussian

A(z = 0, ω) = A0√2πτ exp

∙−12τ 2ω2

¸. (2.82)

Note, in the following we will often use the Gaussian Integral

1√2πσ

∞Z−∞

e−x2

2σ e−jxςdx = e−σ2ς2 for Re σ ≥ 0 (2.83)

In the spectral domain the solution at an arbitray propagation distance z is

A(z, ω) = A0√2πτ exp

∙−12

¡τ 2 + jk00z

¢ω2¸. (2.84)

With the Gaussian integral (2.83) the inverse Fourier transform is

A(z, t0) = A0

µτ 2

(τ 2 + jk00z)

¶1/2exp

∙−12

t02

(τ 2 + jk00z)

¸(2.85)

The exponent can be written as real and imaginary part and we finally obtain

A(z, t0) = A0

µτ 2

(τ 2 + jk00z)

¶1/2exp

"−12

τ 2t02¡τ 4 + (k00z)2

¢ + j12k00z

t02¡τ 4 + (k00z)2

¢#(2.86)

As we see from Eq.(2.86) during propagation the FWHM of the Gaussiandetermined by

exp

"−τ

2(τ 0FWHM/2)2¡τ 4 + (k00z)2

¢ # = 0.5 (2.87)

changes fromτFWHM = 2

√ln 2 τ (2.88)

at the start to

τ 0FWHM = 2√ln 2 τ

s1 +

µk00L

τ 2

¶2(2.89)

= τFWHM

s1 +

µk00L

τ 2

¶2


at z = L. For large stretching this result simplifies to

τ 0FWHM = 2√ln 2

¯k00L

τ

¯for

¯k00L

τ 2

¯À 1. (2.90)

The strongly dispersed pulse has a width equal to the difference in groupdelay over the spectral width of the pulse.Figure 2.11 shows the evolution of the magnitude of the Gaussian wave

packet during propagation in a medium which has no higher order dispersionin normalized units, i.e. ([z] =m, [t] =s, [kj] =s2/m, kj = 2). The pulsespreads continuously.

0

0.5

1

1.5 -6-4

-20

24

6

0.2

0.4

0.6

0.8

1

Ampl

itude

Distance z

Time

Figure 2.11: Magnitude of the complex envelope of a Gaussian pulse,|A(z, t0)| , in a dispersive medium.

As discussed before, the origin of this spreading is the group velocitydispersion (GVD), k00 6= 0. The group velocity varies over the pulse spectrumsignificantly leading to a group delay dispersion (GDD) after a propagationdistance z = L of k00L 6= 0, for the different frequency components. This leadsto the build-up of chirp in the pulse during propagation. We can understandthis chirp by looking at the parabolic phase that develops over the pulse in


time at a fixed propagation distance. The phase is, see Eq.(2.86)

φ(z = L, t0) = −12arctan

∙k00L

τ 2

¸+1

2k00L

t02¡τ 4 + (k00L)2

¢ . (2.91)

(a) Phase

Time t

k'' < 0

k'' > 0

Front Back

Instantaneous Frequency

Time t

k'' < 0

k'' > 0

(b)

Figure 2.12: (a) Phase and (b) instantaneous frequency shift of a Gaussianpulse during propagation through a medium with positive or negative dis-persion.

This parabolic phase, see Fig. 2.12 (a), can be understood as a localyvarying frequency in the pulse, i.e. the derivative of the phase gives theinstantaneous frequency shift in the pulse with respect to the center frequency

ω(z = L, t0) =∂

∂t0φ(L, t0) =

k00L¡τ 4 + (k00L)2

¢t0 (2.92)

see Fig.2.12 (b). The instantaneous frequency indicates that for a mediumwith positive GVD, ie. k00 > 0, the low frequencies are in the front of the


pulse, whereas the high frequencies are in the back of the pulse, since thesub-pulses with lower frequencies travel faster than sub-pulses with higherfrequencies. The opposite is the case for negative dispersive materials.It is instructive for later purposes, that this behaviour can be completely

understood from the center of mass motion of the sub-pulses, see Figure 2.10.Note, we can choose a set of sub-pulses, with such narrow bandwidth, thatdispersion does not matter. In the time domain, these pulses are of coursevery long, because of the time bandwidth relationship. Nevertheless, sincethey all have different carrier frequencies, they interfere with each other insuch a way that the superposition is a very narrow pulse. This interference,becomes destroyed during propagation, since the sub-pulses propagate atdifferent speed, i.e. their center of mass propagates at different speed.

-5

0

5

Tim

e, t

2.01.51.00.50.0

Propagation distance, z

g21/v

k">0 k"<0

Δvg1Δvg2

Δvg3

Figure 2.13: Pulse spreading by following the center of mass of sub-pulsesaccording to Fig. 2.10. For z < 1, the pulses propagate in a medium withpositive dispersion and for z > 1 in a medium with negative dispersion.

The differential group delay ∆Tg(ω) = k00Lω of a sub-pulse with its cen-ter frequency ω different from 0, is due to its differential group velocity∆vg(ω) = −vg0∆Tg(ω)/Tg0 = −v2g0k00ω. Note, that Tg0 = L/vg0. This is illus-trated in Figure 2.13 by ploting the trajectory of the relative motion of thecenter of mass of each sub-pulse as a function of propagation distance, whichasymptotically approaches the formula for the pulse width of the highly dis-


persed pulse Eq.(2.90). If we assume that the pulse propagates through anegative dispersive medium following the positive dispersive medium, thegroup velocity of each sub-pulse is reversed. The sub-pulses propagate to-wards each other until they all meet at one point (focus) to produce againa short and unchirped initial pulse, see Figure 2.13. This is a very powerfultechnique to understand dispersive wave motion and is also the connectionbetween ray optics and physical optics.

2.1.8 Loss and Gain

If the medium considered has loss, described by the imaginary part of thedielectric susceptibility, see (2.44) and Fig. 2.3, we can incorporate this lossinto a complex refractive index

n(Ω) = nr(Ω) + jni(Ω) (2.93)

vian(Ω) =

q1 + eχ(Ω). (2.94)

For an optically thin medium, i.e. eχ¿ 1 the following approximation is veryuseful

n(Ω) ≈ 1 +eχ(Ω)2

. (2.95)

As one can show (in Recitations) the complex susceptibility (2.44) can beapproximated close to resonance, i.e. Ω ≈ Ω0, by the complex Lorentzianlineshape eχ(Ω) = −jχ0

1 + jQΩ−Ω0Ω0

, (2.96)

where χ0 = Qω2p2Ω20

will turn out to be related to the peak absorption of theresonance, which is proportional to the density of atoms. Ω0 is the centerfrequency and ∆Ω = Ω0

Qis the half width half maximum (HWHM) linewidth

of the transition. The real and imaginary part of the complex Lorentzian are

eχr(Ω) =−χ0 (Ω−Ω0)∆Ω

1 +¡Ω−Ω0∆Ω

¢2 , (2.97)

eχi(Ω) =−χ0

1 +¡Ω−Ω0∆Ω

¢2 , . (2.98)


In the derivation of the wave equation for the pulse envelope (2.65) insection 2.1.7, there was no restriction to a real refractive index. Therefore,the wave equation (2.65) also treats the case of a complex refractive index.If we assume a medium with the complex refractive index (2.95), then thewavenumber is given by

K(Ω) =Ω

c0

µ1 +

1

2(eχr(Ω) + jeχi(Ω))¶ . (2.99)

Since we introduced a complex wavenumber, we have to redefine the groupvelocity as the inverse derivative of the real part of the wavenumber withrespect to frequency. At line center, we obtain

υ−1g =∂Kr(Ω)

∂Ω

¯Ω0

=1

c0

µ1− χ0

2

Ω0∆Ω

¶. (2.100)

Thus, for a narrow absorption line, χ0 > 0 and Ω0∆Ω

À 1, the absolute valueof the group velocity can become much larger than the velocity of light invacuum. The opposite is true for an amplifying medium, χ0 < 0. There isnothing wrong with this finding, since the group velocity only describes themotion of the peak of a Gaussian wave packet, which is not a causal wavepacket. A causal wave packet is identical to zero for some earlier time t < t0,in some region of space. A Gaussian wave packet fills the whole space at anytime and can be reconstructed by a Taylor expansion at any time. Therefore,the motion of the peak of such a signal with a speed greater than the speedof light does not contradict special relativity.The imaginary part in the wave vector (2.99) leads with K = Ω

c0to ab-

sorption

α(Ω) = −K2eχi(Ω). (2.101)

In the envelope equation (2.64) for a wavepacket with carrier frequency ω0 =Ω0 and wave number K0 =

Ω0c0the loss leads to a term of the form

∂A(z, ω)

∂z

¯¯(loss)

= −α(Ω0 + ω)A(z, ω) =−χ0K0/2

1 +¡

ω∆Ω

¢2 A(z, ω). (2.102)

In the time domain, we obtain up to second order in the inverse linewidth

∂A(z, t0)

∂z

¯(loss)

= −χ0K0

2

µ1 +

1

∆Ω2∂2

∂t2

¶A(z, t0), (2.103)


which corresponds to a parabolic approximation of the line shape at linecenter, (Fig. 2.3). As we will see later, for an amplifying optical transitionwe obtain a similar equation. We only have to replace the loss by gain

∂A(z, t0)

∂z

¯(gain)

= g

µ1 +

1

Ω2g

∂2

∂t2

¶A(z, t0), (2.104)

where g = −χ0K0

2is the peak gain at line center per unit length and Ωg is

the HWHM linewidth of a transition providing gain.

2.1.9 Kramers-Kroenig Relations and Sellmeier Equa-tion

The linear susceptibility is the frequency response or impulse response of alinear system to an applied electric field, see Eq.(2.42). For a real physicalsystem this response is causal, and therefore real and imaginary parts obeyKramers-Kroenig Relations

χr(Ω) =2

π

∞Z0

ωχi(ω)

ω2 −Ω2dω = n2r(Ω)− 1, (2.105)

χi(Ω) = −2π

∞Z0

Ωχr(ω)

ω2 −Ω2dω. (2.106)

For optical media these relations have the consequence that the refractiveindex and absorption of a medium are not independent, which can oftenbe exploited to compute the index from absorption data or the other wayaround. The Kramers-Kroenig Relations also give us a good understandingof the index variations in transparent media, which means the media are usedin a frequency range far away from resonances. Then the imaginary part ofthe susceptibility related to absorption can be approximated by

χi(Ω) =Xi

Aiδ (ω − ωi) (2.107)

and the Kramers-Kroenig relation results in the Sellmeier Equation for therefractive index

n2(Ω) = 1 +Xi

Aiωi

ω2i −Ω2= 1 +

Xi

aiλ

λ2 − λ2i. (2.108)


This formula is very useful in fitting the refractive index of various mediaover a large frequency range with relatively few coefficients. For exampleTable 2.3 shows the Sellmeier coefficients for fused quartz and sapphire.

Fused Quartz Sapphirea1 0.6961663 1.023798a2 0.4079426 1.058364a3 0.8974794 5.280792λ21 4.679148·10−3 3.77588·10−3λ22 1.3512063·10−2 1.22544·10−2λ23 0.9793400·102 3.213616·102

Table 2.3: Table with Sellmeier coefficients for fused quartz and sapphire.

In general, each absorption line contributes a corresponding index changeto the overall optical characteristics of a material, see Fig. 2.14. A typicalsituation for a material having resonances in the UV and IR, such as glass,is shown in Fig. 2.15. As Fig. 2.15 shows, due to the Lorentzian lineshape, outside of an absorption line the refractive index is always decreasingas a function of wavelength. This behavior is called normal dispersion andthe opposite behavior abnormal dispersion typically only occurs inside ofabsorption lines.

dn

dλ< 0 : normal dispersion (blue refracts more than red)

dn

dλ> 0 : abnormal dispersion

This behavior is also responsible for the mostly positive group delay disper-sion (GVD) or group delay dispersion (GDD) over the transparency range ofa material, which are closely related to dn

dλ.


Figure 2.14: Each absorption line must contribute to an index change viathe Kramers-Kroenig relations.

Figure 2.15: Typcial distribution of absorption lines in a medium transparentin the visible.

Fig. 2.16 shows the transparency range of some often used media.


Figure 2.16: Transparency range of some materials according to [6], p. 175.

GVD and GDD are defined as the variation of the inverse group velocityor group delay as a function of frequency, see Eqs.() , i.e.

GVD =d2k(ω)

dω2

¯ω=0

=d

dω

1

υg(ω)

¯ω=0

(2.109)

GDD =d2k(ω)

dω2

¯ω=0

L =d

dω

L

υg(ω)

¯ω=0

=d

dωTg(ω)|ω=0 (2.110)

where Tg(ω) = L/υg(ω) is the group delay of a wave packet with relativecenter frequency ω. Often theses quantities need to be calculated from therefractive index given by the Sellmeier equation, i.e. n(λ). The correspondingquantities are listed in Table 2.4. The computations are done by substitutingthe frequency with the wavelength.


Dispersion Characteristic Definition Comp. from n(λ)

medium wavelength: λn λn

λn(λ)

medium wavenumber: kn 2πλn

2πλn(λ)

phase velocity: υp ωk

c0n(λ)

group velocity: υg dωdk; dλ = −λ2

2πc0dω c0

n

¡1− λ

ndndλ

¢−1group velocity dispersion: GVD d2k

dω2λ3

2πc20

d2ndλ2

group delay: Tg = Lυg= dφ

dωdφdω= d(kL)

dωnc0

¡1− λ

ndndλ

¢L

group delay dispersion: GDD dTgdω= d2(kL)

dω2λ3

2πc20

d2ndλ2

L

Table 2.4: Table with important dispersion characteristics and how tocompute them from the wavelength dependent refractive index n(λ) using|dω| = 2πc0/λ2d|λ|.


2.2 Electromagnetic Waves and Interfaces

Many microwave and optical devices are based on the characteristics of elec-tromagnetic waves undergoing reflection or transmission at interfaces be-tween media with different electric or magnetic properties characterized byand μ, see Fig. 2.17. Without restriction we can assume that the interfaceis the (x-y-plane) and the plane of incidence is the (x-z-plane). An arbi-trary incident plane wave can always be decomposed into two components.One component is called the transverse electric (TE)-wave or also s-polarizedwave. It has its electric field perpenticular to the plane of incidence. Theother component is called the TM-wave or also p-polarized wave. It has itselectric field in the plane of incidence. The most general case of an incidentmonochromatic TEM-wave is a linear superposition of a TE and a TM-wave.

x

ErEiHr

kr

Hiki

Θi Θr

Θt

z EtHt

kt

Reflection of TE-Wave

x

ErEi

kr

Hiki

Θi Θr

Θt

z Et

Ht

kt

Reflection of TM-Wave

Hr

Figure 2.17: a) Reflection of a TE-wave at an interface, b) Reflection of aTM-wave at an interface

The fields for both cases are summarized in table 2.5

2.2. ELECTROMAGNETIC WAVES AND INTERFACES 45

TE-wave TM-wave

Ei =Ei ej(ωt−ki·r)ey Ei =Ei e

j(ωt−ki·r)ei

Hi =Hi ej(ωt−ki·r)hi Hi =Hi e

j(ωt−ki·r)ey

Er =Er ej(ωt−kr·r)ey Er =Er e

j(ωt−kr·r)r er

Hr =Hr ej(ωt−kr·r)hr Hr =Er e

j(ωt−kr·r)ey

Et =Et ej(ωt−kt·r)ey Et =Et e

j(ωt−kt·r)etHt =Ht e

j(ωt−kt·r)ht Ht =Ht ej(ωt−kt·r)ey

Table 2.5: Electric and magnetic fields for TE- and TM-waves.

with wave vectors of the waves given by

ki = kr = k0√

1μ1,

kt = k0√

2μ2,

ki,t = ki,t (sin θi,t ex + cos θi,t ez) ,

kr = ki (sin θr ex − cos θr ez) ,

and unit vectors defining the directions of the transmitted and reflected mag-netic and electric fields, respectively

hi,t = − cos θi,t ex + sin θi,t ez,hr = cos θr ex + sin θr ez,

ei,t = −hi,t = cos θi,t ex − sin θi,t ez,

er = −hr = − cos θr ex − sin θr ez.

2.2.1 Boundary Conditions and Snell’s law

From 6.013, we know that Stoke’s and Gauss’ Law for the electric and mag-netic fields require constraints on some of the field components at mediaboundaries. In the absence of surface currents and charges, the tangentialelectric and magnetic fields as well as the normal dielectric and magneticfluxes have to be continuous when going from medium 1 into medium 2 forall times at each point along the surface, i.e. the surface z = 0

Ei,x/yej(ωt−ki,xx) +Er,x/ye

j(ωt−kr,xx) = Et,x/yej(ωt−kt,xx). (2.111)

Hi,x/yej(ωt−ki,xx) +Hr,x/ye

j(ωt−kr,xx) = Ht,x/yej(ωt−kt,xx). (2.112)


This equation can only be fulfilled at all times if and only if the x-componentof the k-vectors for the reflected and transmitted wave are equal to (match)the corresponding component of the incident wave

ki,x = kr,x = kt,x (2.113)

This phase matching condition is shown in Fig. 2.18 for the case√

2μ2 >√1μ1 or kt > ki. The semi-circles define the dispersion relation for the waves

of the same frequency in the different media. The x-components of the wavevectors need to be matched.

x

krkiΘi Θr

Θt

z

kt

ε μ1 1,

ε μ2 2 ,

Figure 2.18: Phase matching condition for reflected and transmitted wave

The phase matching condition Eq(2.113) results in θr = θi = θ1 andSnell’s law for the angle θt = θ2 of the transmitted wave

sin θt =

√1μ1√2μ2

sin θi (2.114)

or for the case of non magnetic media with μ1 = μ2 = μ0

sin θt =n1n2sin θi (2.115)


2.2.2 Measuring Refractive Index with a Prism

Snell’s law can be used for measuring the refractive index of materials. Con-sider a prism prepared from a material with unknown refractive index n(λ),see Fig. 2.19 (a).

Figure 2.19: (a) Beam propagating through a prism. (b) For the case ofminimum deviation [3] p. 65.

The prism is mounted on a rotation stage as shown in Fig. 2.20. Theangle of incidence α is then varied with a fixed incident beam path andthe transmitted light is observed on a screen. If one starts of with normalincidence on the first prism surface one notices that after turning the prismone goes through aminium for the deflection angle of the beam. This becomesobvious from Fig. 2.19 (b). There is an angle of incidence α where the beampath through the prism is symmetric. If the input angle is varied around thispoint, it would be identical to exchange the input and output beams. Fromthat we conclude that the deviation δ must go through an extremum at thesymmetry point, see Figure 2.21. It can be shown (Recitations), that therefractive index is then determined by

n =sin γ+δmin

2

sin γ2

, (2.116)

where δmin is the minimum deflection angle. If the measurement is repeatedfor various wavelength of the incident radiation the complete wavelengthdependent refractive index is characterized, see for example, Fig. 2.22.


Figure 2.20: Refraction of a Prism with n=1.731 for different angles of in-cidence alpha. The angle of incidence is stepwise increased by rotating theprism clockwise. The angle of transmission first increases. After the anglefor minimum deviation is reached the transmission angle starts to decrease[3] p67.


Figure 2.21: Deviation versus incident angle [1]

Figure 2.22: Refractive index as a function of wavelength for various mediatransmissive in the visible [1], p42.


2.2.3 Fresnel Reflection

After understanding the direction of the reflected and transmitted light, for-mulas for how much light is reflected and transmitted are derived by eval-uating the boundary conditions for the TE and TM-wave. According toEqs.(2.111) and (2.113) we obtain for the continuity of the tangential E andH fields:

TE-wave (s-pol.) TM-wave (p-pol.)Ei+Er = Et Ei cos θi−Er cos θr =Et cos θtHi cos θi−Hr cos θr =Ht cos θt Hi +Hr = Ht

(2.117)

Introducing the characteristic impedances in both half spaces Z1/2 =q

μ0μ1/2

0 1/2,

and the impedances that relate the tangential electric and magnetic fieldcomponents ZTE/TM

1/2 in both half spaces the boundary conditions can berewritten in terms of the electric or magnetic field components. Note, thatfor complex susceptibilities the wave impedances become complex.

TE-wave (s-pol.) TM-wave (p-pol.)

ZTE1/2 =

Ei/t

Hi/t cos θi/t=

Z1/2cos θ1/2

ZTM1/2 =

Ei/t cos θi/tHi/t

= Z1/2 cos θ1/2

Ei+Er = Et Hi−Hr =ZTM2ZTM1

Ht

Ei−Er =ZTE1ZTE2

Et Hi +Hr = Ht

(2.118)

Amplitude Reflection and Transmission Coefficients

These equations can be easily solve for the amplitude reflection and trans-mission coefficients of the waves. By dividing both equations by the incidentwave amplitudes we obtain for the amplitude reflection and transmission co-effcients. Note, we have defined the reflection and transmission coefficientsin terms of the electric fields for the TE-wave and in terms of the magneticfields for the TM-wave.


rTE = Er

Ei; tTE = Et

EirTM = Hr

Hi; tTM = Ht

Hi

1 + rTE = tTE 1− rTM =ZTM2ZTM1

tTM

1 − rTE = ZTE1ZTE2

tTE 1 + rTM = tTM

(2.119)


Note, that the pair of equations for the amplitude reflection and transmis-sion coefficients are similar for the TE- and TM-wave. Thus the amplitudetransmission and reflection coefficients are

tTE/TM =2

1 +ZTE/TM1/2

ZTE/TM2/1

=2Z

TE/TM2/1

ZTE/TM1 + Z

TE/TM2

(2.120)

rTE/TM =ZTE/TM2/1 − Z

TE/TM1/2

ZTE/TM1 + Z

TE/TM2

(2.121)

Despite the simplicity of these formulas, they describe an enormous wealthof phenomena. To get some insight, consider the case of dielectric losslessmedia characterized by its real refractive indices n1 and n2. Then Eqs.(2.120)and (2.121) for the TE and TM case can be written as


ZTE1/2 =

Z1/2cos θ1/2

= Z0n1/2 cos θ1/2

ZTM1/2 = Z1/2 cos θ1/2 =

Z0n1/2

cos θ1/2

rTE = n1 cos θ1−n2 cos θ2n1 cos θ1+n2 cos θ2

rTM =n2

cos θ2− n1cos θ1

n2cos θ2

+n1

cos θ1

tTE = 2n1 cos θ1n1 cos θ1+n2 cos θ2

tTM =2

n2cos θ2

n2cos θ2

+n1

cos θ1

(2.122)


Figure 2.23 shows the evaluation of Eqs.(2.122) for the case of a reflectionat the interface of air and glass with n2 > n1 and (n1 = 1, n2 = 1.5). For TE-polarized light the reflected light changes sign with respect to the incidentlight (reflection at the optically more dense medium). Note, that this πphase shift always happens for reflection from optically less dense mediaindependent from the angle of incidence. This is not so for TM-polarized lightThe π phase shift only occurs for angles larger than θB, which is called theBrewster angle. So for TM-polarized light the amplitude reflection coefficientis zero at Brewster’s angle. This phenomena will be discussed in more detaillater.

0 15 30 45 60 75 90-1.0-0.8-0.6-0.4-0.20.00.20.40.60.81.01.2

56.3

θB

tTM

tTE

rTM

rTE

Am

plitu

de re

fl. a

nd tr

ansm

. coe

ffici

ents

Incident angle θ1 (deg.)

Figure 2.23: The amplitude coefficients of reflection and transmission as afunction of incident angle. These correspond to external reflection n2 > n1at an air-glas interface (n1 = 1, n2 = 1.5).


This behavior changes drastically if we consider the opposite arrange-ment of media, i.e. we consider the glass-air interface with n1 > n2, seeFigure 2.24. Then the TM-polarized light experiences a π-phase shift uponreflection close to normal incidence. For increasing angle of incidence thisreflection coefficient goes through zero at the Brewster angle θ

0

B differentfrom before. However, for large enough angle of incidence the reflection coef-ficient reaches magnitude 1 and stays there. This phenomenon is called totalinternal reflection and the angle where this occurs first is the critical anglefor total internal reflection, θtot. Total internal reflection will be discussed inmore detail later.

0 15 30 45

-0.20.00.20.40.60.81.01.21.41.61.82.0

33.7

θB

41.8

θtot

rTE

rTM

tTE

tTM

Am

plitu

de re

fl. a

nd tr

ansm

. coe

ffici

ents

Incident angle θ1 (deg.)

Figure 2.24: The amplitude coefficients of reflection and transmission as afunction of incident angle. These correspond to internal reflection n1 > n2at a glas-air interface (n1 = 1.5, n2 = 1).


Power reflection and transmission coefficients

Often we are not interested in the amplitude but rather in the optical powerreflected or transmitted in a beam of finite size, see Figure 2.25.

Figure 2.25: Reflection and transmission of an incident beam of finite size[1].

Note, that to get the power in a beam of finite size, we need to integratedthe corresponding Poynting vector over the beam area, which means multi-plication by the beam crosssectional area for a homogenous beam. Since theangle of incidence and reflection are equal, θi = θr = θ1 this beam crosssec-tional area drops out in reflection

RTE/TM =ITE/TMr A cos θi

ITE/TMi A cos θr

=¯rTE/TM

¯2=

¯¯ZTE/TM

2 − ZTE/TM1

ZTE/TM1 + Z

TE/TM2

¯¯2

(2.123)

However, due to the different angles for the incident and the transmittedbeam θt = θ2 6= θ1, we arrive at

T TE/TM =ITE/TMt A cos θt

ITE/TMi A cos θi

(2.124)

=cos θ2cos θ1

Re

(1

Z2/1

)Re

(1

Z1/2

)−1 ¯tTE/TM

¯2.


Using in the case of TE-polarizationZ1/2cos θ1/2

= ZTE1/2 and analogously for TM-

polarization Z1/2 cos θ1/2 = ZTM1/2 , we obtain

T TE/TM = Re

(1

ZTE/TM1/2

)−1Re

⎧⎪⎨⎪⎩ 4ZTE/TM2/1¯

ZTE/TM1 + Z

TE/TM2

¯2⎫⎪⎬⎪⎭ (2.125)

Note, for the case where the characteristic impedances are complex this cannot be further simplified. If the characteristic impedances are real, i.e. themedia are lossless, the transmission coefficient simplifies to

T TE/TM =4Z

TE/TM1/2 Z

TE/TM2/1³

ZTE/TM1 + Z

TE/TM2

´2.

(2.126)

To summarize for lossless media the power reflection and transmission coef-ficients are


ZTE1/2 =

Z1/2cos θ1/2

= Z0n1/2 cos θ1/2

ZTM1/2 = Z1/2 cos θ1/2 =

Z0n1/2

cos θ1/2

RTE =¯n2 cos θ2−n1 cos θ1n1 cos θ1+n2 cos θ2

¯2RTM =

¯n2


n2cos θ2

+n1

cos θ1

¯2T TE = 4n1 cos θ1n2 cos θ2

|n1 cos θ1+n2 cos θ2|2T TM =

4n2

cos θ2

n1cos θ1¯

n2cos θ2

+n1

cos θ1

¯2T TE +RTE = 1 T TM +RTM = 1

(2.127)

A few phenomena that occur upon reflection at surfaces between differentmedia are especially noteworthy and need a more indepth discussion becausethey enhance or enable the construction of many optical components anddevices.

2.2.4 Brewster’s Angle

As Figures 2.23 and 2.24 already show, for light polarized parallel to theplane of incidence, p-polarized light, the reflection coefficient vanishes at agiven angle θB, called the Brewster angle. Using Snell’s Law Eq.(2.115),

n2n1=sin θ1sin θ2

, (2.128)


we can rewrite the reflection and transmission coefficients in Eq.(2.127) onlyin terms of the angles. For example, we find for the reflection coefficient

RTM =

¯¯ n2n1 − cos θ2

cos θ1n2n1+ cos θ2

cos θ1

¯¯2

=

¯¯ sin θ1sin θ2

− cos θ2cos θ1

sin θ1sin θ2

+ cos θ2cos θ1

¯¯2

(2.129)

=

¯sin 2θ1 − sin 2θ2sin 2θ1 + sin 2θ2

¯2(2.130)

where we used in the last step in addition the relation sin 2α = 2 sinα cosα.Thus by forcing RTM = 0, the Brewster angle is reached for

sin 2θ1,B − sin 2θ2,B = 0 (2.131)

or2θ1,B = π − 2θ2,B or θ1,B + θ2,B =

π

2(2.132)

This relation is illustrated in Figure 2.26. The reflected and transmittedbeams are orthogonal to each other, so that the dipoles induced in themedium by the transmitted beam, shown as arrows in Fig. 2.26, can notradiate into the direction of the reflected beam. This is the physical originof the zero in the reflection coefficient, only possible for a p-polarized orTM-wave.The relation (2.132) can be used to express the Brewster angle as a func-

tion of the refractive indices, because if we substitute (2.132) into Snell’s lawwe obtain

sin θ1sin θ2

=n2n1

sin θ1,B

sin¡π2− θ1,B

¢ =sin θ1,Bcos θ1,B

= tan θ1,B,

ortan θ1,B =

n2n1

. (2.133)

Using the Brewster angle condition one can insert an optical component witha refractive index n 6= 1 into a TM-polarized beam in air without havingreflections, see Figure 2.27. Note, this is not possible for a TE-polarizedbeam.


x

Ei Er

krki

z Etkt

Reflection of TM-Wave at Brewster’s Angle

θ1,Β

θ2,Β

θ1,Β

Figure 2.26: Conditions for reflection of a TM-Wave at Brewster’s angle.The reflected and transmitted beams are orthogonal to each other, so thatthe dipoles excited in the medium by the transmitted beam can not radiateinto the direction of the reflected beam.

Figure 2.27: A plate under Brewster’s angle does not reflect TM-light. Theplate can be used as a window to introduce gas filled tubes into a laser beamwithout insertion loss (ideally), [6] p. 209.


2.2.5 Total Internal Reflection

Another striking phenomenon, see Figure 2.24, occurs for the case wherethe beam hits the surface from the side of the optically denser medium, i.e.n1 > n2. There is a critical angle of incidence, beyond which all light isreflected. How can that occur? This is easy to understand from the phasematching diagram at the surface, see Figure 2.18, which is redrawn for thiscase in Figure 2.28.

Figure 2.28: Phase matching diagram for total internal reflection.

There is no real wavenumber in medium 2 possible as soon as the angle ofincidence becomes larger than the critical angle for total internal reflection

θi > θtot (2.134)

withsin θtot =

n2n1

. (2.135)

Figure 2.29 shows the angle of refraction and incidence for the two casesof external (n1 < n2) and internal reflection (n1 > n2), when the angle ofincidence approaches the critical angle.


Figure 2.30: Relation between angle of refraction and incidence for externalrefraction and internal refraction ([1], p. 81).

Figure 2.29: Relation between angle of refraction and incidence for externalrefraction and internal refraction ([6], p. 11).

Total internal reflection enables broadband reflectors. Figure 2.30 shows


again what happens when the critical angle of reflection is surpassed. Fig-ure 2.31 shows how total internal reflection can be used to guide light viareflection at a prism or by multiple reflections in a waveguide.

Figure 2.31: (a) Total internal reflection, (b) internal reflection in a prism,(c) Rays are guided by total internal reflection from the internal surface ofan optical fiber ([6] p. 11).

Figure 2.32 shows the realization of a retro reflector, which always returnsa parallel beam independent of the orientation of the prism (in fact the prismcan be a real 3D-corner so that the beam is reflected parallel independentfrom the precise orientation of the corner cube). A surface patterned by littlecorner cubes constitute a "cats eye" used on traffic signs.

Figure 2.32: Total internal reflection in a retro reflector.

More on reflecting prisms and its use can be found in [1], pages 131-136.


Evanescent Waves

What is the field in medium 2 when total internal reflection occurs? Is itidentical to zero? It turns out phase matching can still occur if the propaga-tion constant in z-direction becomes imaginary, k2z = −jκ2z, because then wecan fulfill the wave equation in medium 2. This is equivalent to the dispersionrelation

k22x + k22z = k22,

or with k2x = k1x = k1 sin θ1, we obtain for the imaginary wavenumber

κ2z =qk21 sin

2 θ1 − k22, (2.136)

= k1psin2 θ1 − sin2 θtot. (2.137)

The electric field in medium 2 is then, for the example for a TE-wave, givenby

Et = Et ey ej(ωt−kt·r), (2.138)

Et ey ej(ωt−k2,xx)e−κ2zz. (2.139)

Thus the wave penetrates into medium 2 exponentially with a 1/e-depth δ,given by

δ =1

κ2z=

1

k1psin2 θ1 − sin2 θtot

(2.140)

Figure 2.33 shows the penetration depth as a function of angle of incidencefor a silica/air interface and a silicon/air interface. The figure demonstratesthat light from inside a semiconductor material with a relatively high indexaround n=3.5 is mostly captured in the semiconductor material (Problem oflight extraction from light emitting diodes (LEDs)), see problem set 3.


0.4

0.2

0.0Pene

tratio

nde

pth,

μm

806040200Angle of incidence, o

n1=3.5n2=1

n1=1.45n2=1

Figure 2.33: Penetration depth for total internal reflection at a silica/air anda silicon/air interface for λ = 1550nm.

As the magnitude of the reflection coefficient is 1 for total internal re-flection, the power flowing into medium 2 must vanish, i.e. the transmissionis zero. Note, that the transmission and reflection coefficients in Eq.(2.122)can be used beyond the critical angle for total internal reflection. We onlyhave to be aware that the electric field in medium 2 has an imaginary depen-dence in the exponent for the z-direction, i.e. k2z = k2 cos θ2 = −jκ2z. Thuscos θ2 in all formulas for the reflection and transmission coefficients has to bereplaced by the imaginary number

cos θ2 =k2zk2= −jk1

k2

psin2 θ1 − sin2 θtot (2.141)

= −jn1n2

psin2 θ1 − sin2 θtot

= −j

sµsin θ1sin θtot

¶2− 1.


Then the reflection coefficients in Eq.(2.122) change to all-pass functions


rTE = n1 cos θ1−n2 cos θ2n1 cos θ1+n2 cos θ2

rTM =n2


n2cos θ2

+n1

cos θ1

rTE =cos θ1+j

n2n1

r³sin θ1sin θtot

´2−1

cos θ1−jn2n1

r³sin θ1sin θtot

´2−1

rTM =cos θ1+j

n1n2

r³sin θ1sin θtot

´2−1

cos θ1−jn1n2

r³sin θ1sin θtot

´2−1

tan φTE

2= 1

cos θ1

n2n1

r³sin θ1sin θtot

´2− 1 tan φTM

2= 1

cos θ1

n1n2

r³sin θ1sin θtot

´2− 1(2.142)

Thus the magnitude of the reflection coefficient is 1. However, there isa non-vanishing phase shift for the light field upon total internal reflection,denoted as φTE and φTM in the table above. Figure 2.34 shows these phaseshifts for the glass/air interface and for both polarizations.

200

100

0

Pha

se in

Ref

lect

ion,

o

806040200Angle of incidence, o

n1=1.45n2=1

TE-Wave (s-polarized)

TM-Wave (p-polarized)

Brewsterangle

Figure 2.34: Phase shifts for TE- and TM- wave upon reflection from asilica/air interface, with n1 = 1.45 and n2 = 1.

Goos-Haenchen-Shift

So far, we looked only at plane waves undergoing reflection at surface due tototal internal reflection. If a beam of finite transverse size is reflected from


such a surface it turns out that it gets displaced by a distance ∆z, see Figure2.35 (a), called Goos-Haenchen-Shift.

Figure 2.35: (a) Goos-Haenchen Shift and related beam displacement uponreflection of a beam with finite size; (b) Accumulation of phase shifts in awaveguide. (c) Goos-Haenchen Shift by reflection at a virtual layer locatedat the penetration depth.

Detailed calculations show, that the displacement is given by

∆z = 2δTE/TM tan θ1, (2.143)

as if the beam was reflected at a virtual layer with depth δTE/TM into medium


2. It turns out, that for TE-waves

δTE = δ, (2.144)

where δ is the penetration depth according to Eq.(2.140) for evanescentwaves. But for TM-waves

δTM =δ∙

1 +³n1n2

´2¸sin2 θ1 − 1

(2.145)

These shifts accumulate when the beam is propagating in a waveguide, seeFigure 2.35 (b) and is important to understand the dispersion relations ofwaveguide modes. The Goos-Haenchen shift can be observed by reflection ata prism partially coated with a silver film, see Figure 2.36. The part reflectedfrom the silver film is shifted with respect to the beam reflected due to totalinternal reflection, as shown in the figure.

Figure 2.36: Experimental proof of the Goos-Haenchen shift by total in-ternal reflection at a prism, that is partially coated with silver, where thepenetration of light can be neglected. [3] p. 486.

Frustrated total internal reflection

Another proof for the penetration of light into medium 2 in the case oftotal internal reflection can be achieved by putting two prisms, where total


internal reflection occurs back to back, see Figure 2.37. Then part of thelight, depending on the distance between the two interfaces, is convertedback into a propagating wave that can leave the second prism. This effect iscalled frustrated internal reflection and it can be used as a beam splitter asshown in Figure 2.37.

Figure 2.37: Frustrated total internal reflection. Part of the light is pickedup by the second surface and converted into a propagating wave.

2.3 Polarization of Electromagnetic Waves

So far we have discussed linearly polarized electromagnetic waves, where theelectric field of a TEM-wave propagating along the z−direction was eitherpolarized along the x− or y−axis. The most general TEM-wave has simul-taneously electric fields in both polarizations and the direction of the electricfield in space, i.e. its polarization, can change during propagation. A de-scription of polarization and polarization evolution in optical systems can beaccomplished using Jones vectors and matrices.

2.3.1 Polarization

A general complex TEM-wave propagating along the z−direction is given by

E(z, t) =

⎛⎝ E0x

E0y

0

⎞⎠ ej(ωt−kz), (2.146)

2.3. POLARIZATION OF ELECTROMAGNETIC WAVES 67

where E0x = E0xejϕx and E0y = E0ye

jϕy are the complex field amplitudes ofthe x− and y− polarized components of the wave. The real electric field isgiven by

E(z, t) =

⎛⎝ E0x cos (ωt− kz + ϕx)E0y cos

¡ωt− kz + ϕy

¢0

⎞⎠ , (2.147)

Both components are periodic functions in ωt− kz = ω (t− z/c) .

Linear Polarization

If the phases of the complex field amplitudes along the x− and y−axis areequal, i.e.

E0x = E0xejϕ and E0y = E0ye

jϕ

then the real electric field

E(z, t) =

⎛⎝ E0xE0y0

⎞⎠ cos (ωt− kz + ϕ) (2.148)

always oscillates along a fixed direction in the x-y-plane, see Figure 2.38

Figure 2.38: Linearly polarized light. (a) Time course at a fixed position z.(b) A snapshot at a fixed time t, [6], p. 197.

The angle between the polarization direction and the x-axis, α, is givenby α = arctan (E0y/E0x) . If there is a phase difference of the complex fieldamplitudes along the x− and y−axis, the direction and magnitude of theelectric field amplitude changes periodically in time at a given position z.


Circular Polarization

Special cases occur when the magnitude of the fields in both linear polariza-tions are equal E0x = E0y = E0, but there is a phase difference ∆ϕ = ±π

2in

both components. Then we obtain

E(z, t) = E0Re

⎧⎨⎩⎛⎝ ejϕ

ej(ϕ−∆ϕ)

0

⎞⎠ ej(ωt−kz)

⎫⎬⎭ (2.149)

= E0

⎛⎝ cos (ωt− kz + ϕ)∓ sin (ωt− kz + ϕ)

0

⎞⎠ . (2.150)

For this case, the tip of the electric field vector describes a circle in thex− y−plane, as

|Ex(z, t)|2 + |Ey(z, t)|2 = E20 for all z, t, (2.151)

see Figure 2.39.

Figure 2.39: Trajectories of the tip of the electric field vector of a right andleft circularly polarized plane wave. (a) Time course at a fixed position z.(b) A snapshot at a fixed time t. Note, the sense of rotation in (a) is oppositeto that in (b) [6], p. 197.


Right Circular Polarization If the tip of the electric field at a given time,t, rotates counter clockwise with respect to the phase fronts of the wave, herein the positive z−direction, then the wave is called right circularly polarizedlight, i.e.

Erc(z, t) = E0Re

⎧⎨⎩⎛⎝ 1

j0

⎞⎠ ej(ωt−kz+ϕ)

⎫⎬⎭ = E0

⎛⎝ cos (ωt− kz + ϕ)− sin (ωt− kz + ϕ)

0

⎞⎠ .

(2.152)A snapshot of the lines traced by the end points of the electric-field vec-

tors at different positions is a right-handed helix, like a right-handed screwpointing in the direction of the phase fronts of the wave, i.e. k−vector seeFigure 2.39 (b).

Left Circular Polarization If the tip of the electric field at a given fixedtime, t, rotates clockwise with respect to the phase fronts of the wave, herein the again in the positive z−direction, then the wave is called left circularlypolarized light, i.e.

Elc(z, t) = E0Re

⎧⎨⎩⎛⎝ 1−j0

⎞⎠ ej(ωt−kz+ϕ)

⎫⎬⎭ = E0

⎛⎝ cos (ωt− kz + ϕ)sin (ωt− kz + ϕ)

0

⎞⎠ .

(2.153)

Eliptical Polarization The general polarization case is called elipticalpolarization, as for arbitrary E0x = E0xe

jϕx and E0y = E0yejϕy , we obtain

for the locus of the tip of the electric field vector from

E(z, t) =

⎛⎝ E0× cos (ωt− kz + ϕx)E0y cos

¡ωt− kz + ϕy

¢0

⎞⎠ . (2.154)

the relations

Ey

E0y= cos

¡ωt− kz + ϕy

¢= cos

¡ωt− kz + ϕx +

¡ϕy − ϕx

¢¢(2.155)

= cos (ωt− kz + ϕx) cos¡ϕy − ϕx

¢(2.156)

− sin (ωt− kz + ϕx) sin¡ϕy − ϕx

¢.


and

Ex

E0x= cos (ωt− kz + ϕx) . (2.157)

These relations can be combined toEy

E0y− Ex

E0xcos¡ϕy − ϕx

¢= − sin (ωt− kz + ϕx) sin

¡ϕy − ϕx

¢(2.158)

sin (ωt− kz + ϕx) =

s1−

µEx

E0x

¶2(2.159)

Substituting Eq.(2.159) in Eq.(2.158) and building the square results in

µEy

E0y− Ex

E0xcos¡ϕy − ϕx

¢¶2=

Ã1−

µEx

E0x

¶2!sin2

¡ϕy − ϕx

¢. (2.160)

After reordering of the terms we obtainµEx

E0x

¶2+

µEy

E0y

¶2− 2 Ex

E0x

Ey

E0ycos¡ϕy − ϕx

¢= sin2

¡ϕy − ϕx

¢. (2.161)

This is the equation of an ellipse making an angle α with respect to the x-axisgiven by

tan 2α =2E0xE0y cos

¡ϕy − ϕx

¢E20x −E2

0y

. (2.162)

see Figure 2.40.

Figure 2.40: (a) Rotation of the endpoint of the electric field vector in thex-y-plane at a fixed position z. (b) A snapshot at a fixed time t [6], p. 197.


Elliptically polarized light can also be understood as a superposition of aright and left cicular polarized light, see Figure 2.41.

Figure 2.41: Elliptically polarized light as a superposition of right and leftcircularly polarized light [1], p. 223.

2.3.2 Jones Calculus

As seen in the last section, the information about polarization of a TEM-wavecan be tracked by a vector that is proportional to the complex electric-fieldvector. This vector is called the Jones vectorµ

E0x

E0y

¶∼ V =

µV x

V y

¶: Jones Vector (2.163)

Jones Matrix

Figure 2.42 shows a light beam that is normally incident on a retardationplate along the z−axis with a polarization state described by a Jones vector.A retardation plate is made up of a crystaline material, such as for exampleQuartz, which has unisotropic dielectric behavior. For example, it showsdifferent refractive index depending on the alignment of the polarization withthe crystal axes. For retardation plates, there are two principle axis, s− forslow and f− for fast axis, i.e. the refractive index along the slow axis islarger than along the fast axis. Let ns and nf be the refractive index of theslow and fast principle axis, respectively, then ns > nf .


Figure 2.42: A retardation plate rotated at an angle ψ about the z-axis.f("fast") and s("slow") are the two principal dielectric axes of the crystal forlight propagating along the z−axis [2], p. 17.

Usually, the principle axis (s− and f− axis) of the retardation plate arerotated by an angle ψ with respect to the x− and y−axis. The polarizationstate of the emerging beam in the crystal coordinate system is thus given byµ

V 0s

V 0f

¶=

µe−jkonsL 00 e−jkonfL

¶µVsVf

¶, (2.164)

The phase retardation is defined as the phase difference between the twocomponents

Γ = (ns − nf) koL. (2.165)

In birefringent crystals the difference in refractive index is much smallerthan the index itself, |ns − nf | ¿ ns, nf , therefore parallel to the evolvingdifferential phase a large absolute phase shift occurs. Taking the mean phaseshift

φ =1

2(ns + nf) koL, (2.166)

out, we can rewrite (2.164) asµV 0s

V 0f

¶= e−jφ

µe−jΓ/2 00 ejΓ/2

¶µVsVf

¶. (2.167)


The matrix connecting the Jones vector at the input of an optical componentwith the Jones vector at the output is called a Jones matrix.If no coherent additon with another field is planned at the output of the

system, the average phase φ can be dropped. With the rotation matrix, R,connecting the (x, y) coordinate system with the (s, f) coordinate system

R (ψ) =

µcosψ sinψ− sinψ cosψ

¶, (2.168)

we find the Jones matrix W describing the propagation of the field compo-nents through the retardation plate asµ

V 0x

V 0y

¶=W

µVxVy

¶. (2.169)

withW = R (−ψ)W0R (ψ) . (2.170)

and

W0 =

µe−jΓ/2 00 ejΓ/2

¶. (2.171)

Carrying out the matix multiplications leads to

W =

µe−jΓ/2 cos2(ψ) + ejΓ/2 sin2(ψ) −j sin Γ

2sin (2ψ)

−j sin Γ2sin (2ψ) e−jΓ/2 sin2(ψ) + ejΓ/2 cos2(ψ)

¶.

(2.172)Note that the Jones matrix of a wave plate is a unitary matrix, that is

W †W = 1.

Unitary matrices have the property that they transform orthogonal vectorsinto another pair of orthogonal vectors. Thus two orthogonal polarizationstates remain orthogonal when propagating through wave plates.

Polarizer

A polarizer is a device that absorbs one component of the polarization vector.The Jones matrix of polarizer along the x-axis or y-axis is

Px =

µ1 00 0

¶, and Py =

µ0 00 1

¶. (2.173)


Half-Wave Plate

A half-wave plate has a phase retardation of Γ = π, i.e. its thickness ist = λ/2(ne − no). The corresponding Jones matrix follows from Eq.(2.172)

W = −jµcos(2ψ) sin (2ψ)sin (2ψ) − cos(2ψ)

¶. (2.174)

For the special case of ψ = 45o, see Figure 2.43, the half-wave plate rotatesa linearly polarized beam exactly by 900, i.e. it exchanges the polarizationaxis. It can be shown, that for a general azimuth angle ψ, the half-waveplate will rotate the polarization by an angle 2ψ, see problem set. Whenthe incident light is circularly polarized a half-wave plate will convert right-hand circularly polarized light into left-hand circularly polarized light andvice versa, regardless of the azimuth angle ψ.

Figure 2.43: The effect of a half-wave plate on the polarziation state of abeam, [2], p.21.

Quarter-Wave Plate

A quarter-wave plate has a phase retardation of Γ = π/2, i.e. its thicknessis t = λ/4(ne − no). The corresponding Jones matrix follows again fromEq.(2.172)


W =

Ã1√2[1− j cos(2ψ)] −j 1√

2sin (2ψ)

−j 1√2sin (2ψ) 1√

2[1 + j cos(2ψ)]

!. (2.175)

and for the special case of ψ = 45o, see Figure 2.43 we obtain

W =1√2

µ1 −j−j 1

¶, (2.176)

see Figure 2.44.

Figure 2.44: The effect of a quarter wave plate on the polarization state of alinearly polarized input wave [2], p.22.

If the incident beam is vertically polarized, i.e.µVxVy

¶=

µ01

¶, (2.177)

the effect of a 45o -oriented quarter-wave plate is to convert vertically polar-ized light into right-handed circularly polarized light.µ

V 0x

V 0y

¶=−j√2

µ1j

¶. (2.178)


If the incident beam is horizontally polarizedµVxVy

¶=

µ10

¶, (2.179)

the outgoing beam is a left-handed circularly polarized, see Figure 2.44.µV 0x

V 0y

¶=

1√2

µ1−j

¶. (2.180)

2.4 Interference and Interferometers

A phenomenon characteristic for waves is interference. Many optical devicesare based on the concept of interfering waves, such as interferometers, lowloss dielectric mirrors and other multilayer optical coatings. After having aquick look into the phenomenon of interference, we will develope a powerfulmatrix formalism that enables us to evaluate efficiently many optical (alsomicrowave) systems based on interference.

2.4.1 Interference and Coherence

Interference

Interference of waves is a consequence of the linearity of the wave equation(2.7). If we have two individual solutions of the wave equation

E1(r, t) = E1 cos(ω1t− k1 · r + ϕ1) e1, (2.181)

E2(r, t) = E2 cos(ω2t− k1 · r + ϕ2) e2, (2.182)

with arbitrary amplitudes, wave vectors and polarizations, the sum of thetwo fields (superposition) is again a solution of the wave equation

E(r, t) = E1(r, t) +E2(r, t). (2.183)

If we look at the intensity, which is proportional to the amplitude square ofthe total field

E(r, t)2 =³E1(r, t) + E2(r, t)

´2, (2.184)

2.4. INTERFERENCE AND INTERFEROMETERS 77

we find

E(r, t)2 = E1(r, t)2 +E2(r, t)

2 + 2E1(r, t) ·E2(r, t) (2.185)

with

E1(r, t)2 =

E21

2

³1 + cos 2(ω1t− k1 · r + ϕ1)

´, (2.186)

E2(r, t)2 =

E22

2

³1 + cos 2(ω2t− k2 · r + ϕ2)

´, (2.187)

E1(r, t) · E2(r, t) = (e1 · e2)E1E2 cos(ω1t− k1 · r + ϕ1) · (2.188)

· cos(ω2t− k2 · r + ϕ2)

E1(r, t) ·E2(r, t) =1

2(e1 · e2)E1E2 · (2.189)

·

⎡⎣ cos³(ω1 − ω2) t−

³k1 − k2

´· r + (ϕ1 − ϕ2)

´+cos

³(ω1 + ω2) t−

³k1 + k2

´· r + (ϕ1 + ϕ2)

´ ⎤⎦ (2.190)

Since at optical frequencies neither our eyes nor photo detectors, can everfollow the optical frequency itself and certainly not twice as large frequencies,we drop the rapidly oscillating terms. Or in other words we look only on thecycle-averaged intensity, which we denote by a bar

E(r, t)2 =E21

2+

E22

2+ (e1 · e2)E1E2 ·

· cos³(ω1 − ω2) t−

³k1 − k2

´· r + (ϕ1 − ϕ2)

´(2.191)

Depending on the frequencies ω1 and ω2 and the deterministic and stochasticproperties of the phases ϕ1 and ϕ2, we can detect this periodically varyingintensity pattern called interference pattern. Interference of waves can bebest visuallized with water waves, see Figure 2.45. Note, however, that waterwaves are a scalar field, whereas the EM-waves are vector waves. Therefore,the interference phenomena of EM-waves are much richer in nature thanfor water waves. Notice, from Eq.(2.191), it follows immediately that theinterference vanishes in the case of orthogonally polarized EM-waves, becauseof the scalar product involved. Also, if the frequencies of the waves are notidentical, the interference pattern will not be stationary in time.


Figure 2.45: Interference of water waves from two point sources in a rippletank [1] p. 276.

If the frequencies are identical, the interference pattern depends on thewave vectors, see Figure 2.46.

k1

k2

Wavefronts

Lines of constant differentialphase

Λ

Figure 2.46: Interference pattern generated by two monochromatic planewaves.


The interference pattern has itself a wavevector given by

k1 − k2 (2.192)

and therefore varies periodically with period

Λ =2π¯

k1 − k2

¯ . (2.193)

Coherence

The ability of waves to generate an interference pattern is called coherence.Coherence can be quantified both temporally or spatially. For example, if weare at a certain position r in the interference pattern described by Eq.(2.191),we will only have stationary conditions over a time interval

Tcoh <<2π

ω1 − ω2.

Thus the spectral width of the waves determines the temporal coherence.However, it depends very often on the expermental arrangement whether agiven situation can still lead to interference or not. Even so the interferinglight may be perfectly temporally coherent, i.e. perfectly monochromatic,ω1 = ω2,yet the wave vectors may not be stable over time and the spatialinteference pattern may wash out, i.e. there is insufficient spatial coherence.So for stable and maximum interference three conditions must be fulfilled:

• stable and identical polarization

• small change in the relative phase between the beams involved over theobservation time, temporal coherence, often achieved by using narrrowlinewidth light

• stable beam propagation or guiding of light in a waveguide to achievespatial coherence.

It is by no means trivial to arrive at a light source and an experimen-tal setup that enables good coherence and strong interference of the beamsinvolved.Interference of beams can be used to measure relative phase shifts between

them which may be proportional to a physical quantity that needs to be


measured. Such phase shifts between two beams can also be used to modulatethe light output at a given position in space via interference. In 6.013, wehave already encountered interference effects between forward and backwardtraveling waves on transmission lines. This is very closely related to what weuse in optics, therefore, we quickly relate the TEM-wave progagation to thetransmission line formalism developed in Chapter 5 of 6.013.

2.4.2 TEM-Transmission Lines and TEM-Waves

The voltage V and current I along a TEM transmission line with an induc-tance L0 and a capacitance C0 per unit length satisfy the following set ofdifferential equations also called transmission line equations

∂V (t, z)

∂z= −L0∂I(t, z)

∂t(2.194)

∂I(t, z)

∂z= −C 0∂V (t, z)

∂t(2.195)

Substitution of these equations into each other results in wave equations foreither the voltage or the current

∂2V (t, z)

∂z2− 1

c2∂2V (t, z)

∂t2= 0, (2.196)

∂2I(t, z)

∂z2− 1

c2∂2I(t, z)

∂t2= 0, (2.197)

where c = 1/√L0C 0 is the speed of wave propagation on the transmission

line. The ratio between voltage and current for monochomatic waves is thecharacteristic impedance Z =

pL0/C 0.

The equations of motion for the electric and magnetic field of a x-polarizedTEM wave according to Figure 2.1, with E−field along the x-axis and H-fields along the y- axis follow directly from Faraday’s and Ampere’s law

∂E(t, z)

∂z= −μ∂H(t, z)

∂t, (2.198)

∂H(t, z)

∂z= −ε∂E(t, z)

∂t, (2.199)

which are identical to the transmission line equations (2.194) and (2.195).Substitution of these equations into each other results again in wave equations


for electric and magnetic fields propagating at the speed of light c = 1/√με

and with characteristic impedance ZF =pμ/ε. Thus the tansmission line

equations describe the same phenomena than a one dimensional TEM-wave.The solutions of the wave equation are forward and backward traveling

waves, which can be decoupled by transforming the fields to the forward andbackward traveling waves

a(t, z) =1

2

rAeff

ZF(E(t, z) + ZFH(t, z)) , (2.200)

b(t, z) =1

2

rAeff

ZF(E(t, z)− ZFH(t, z)) , (2.201)

which fulfill the equationsµ∂

∂z+1

c

∂

∂t

¶a(t, z) = 0, (2.202)µ

∂

∂z− 1

c

∂

∂t

¶b(t, z) = 0. (2.203)

For a homogeneous medium or transmission line the equations of the for-ward and backward traveling waves are uncoupled. Then the dynamics isvery simple. Note, we introduced that cross section Aeff such that |a|2 isproportional to the total power carried by the wave. Clearly, the solutionsare

a(t, z) = f(t− z/c0), (2.204)

b(t, z) = g(t+ z/c0), (2.205)

which resembles the D’Alembert solutions of the wave equations for the elec-tric and magnetic field

E(t, z) =

sZFo

Aeff(a(t, z) + b(t, z)) , (2.206)

H(t, z) =

s1

2ZFoAeff(a(t, z)− b(t, z)) . (2.207)

Here, the forward and backward propagating fields are already normalizedsuch that the Poynting vector multiplied with the effective area gives alreadythe total power transported by the fields in the effective cross section Aeff

P = S · (Aeff ez) = AeffE(t, z)H(t, z) = |a(t, z)|2 − |b(t, z)|2 . (2.208)


In 6.013, it was shown that the relation between sinusoidal current andvoltage waves

V (t, z) = Re©V (z)ejωt

ªand I(t, z) = Re

©I(z)ejωt

ª(2.209)

along the transmission line or corresponding electric and magnetic fields inone dimensional wave propagation is described by a generalized compleximpedance Z(z) that obey’s certain transformation rules, see Figure 2.47(a).

Figure 2.47: (a) Transformation of generalized impedance along transmissionlines, (b) Transformation of generalized impedance accross free space sectionswith different characterisitc wave impedances in each section.

Along the first transmission line, which is terminated by a load impedance,the generalized impedance transforms according to

Z1(z) = Z1 ·Z0 − jZ1 tan (k1z)

Z1 − jZ0 tan (k1z)(2.210)

with k1 = k0n1 and along the second transmission line the same rule appliesas an example

Z2(z) = Z2 ·Z1(−L1)− jZ2 tan (k2z)

Z2 − jZ1(−L1) tan (k2z)(2.211)


with k2 = k0n2. Note, that the media can also be lossy, then the character-istic impedances of the transmission lines and the propagation constants arealready themselves complex numbers. The same formalism can be used tosolve corresponding one dimensional EM-wave propagation problems.

Antireflection Coating

The task of an antireflection (AR-)coating, analogous to load matching intransmission line theory, is to avoid reflections between the interface of twomedia with different optical properties. One method of course could be toplace the interface at Brewster’s angle. However, this is not always possible.Let’s assume we want to put a medium with index n into a beam undernormal incidence, without having reflections on the air/medium interface.The medium can be for example a lens. This is exactly the situation shownin Figure 2.47 (b). Z2 describes the refractive index of the lense material,e.g. n2 = 3.5 for a silicon lense, we can deposit on the lens a thin layerof material with index n1 corresponding to Z1 and this layer should matchto the free space index n0 = 1 or impedance Z0 = 377Ω. Using (2.210) weobtain

Z2 = Z1(−L1) = Z1Z0 − jZ1 tan (−k1L1)Z1 − jZ0 tan (−k1L1)

(2.212)

If we choose a quarter wave thick matching layer k1L1 = π/2, this simplifiesto the important result

Z2 =Z21Z0, with Z1 =

Z0n1and Z2 =

Z0n2

(2.213)

or n1 =√n2n0 and L1 =

λ

4n1. (2.214)

Thus a quarter wave AR-coating needs a material which has an index cor-responding to the geometric mean of the two media to be matched. In thecurrent example this would be n2 =

√3.5 ≈ 1.87

2.4.3 Scattering and Transfer Matrix

Another formalism to analyze optical systems (or microwave circuits) canbe formulated using the forward and backward propagating waves, whichtransform much simpler along a homogenous transmission line than the total


fields, i.e. the sum of forward and backward waves. However, at interfacesscattering of these waves occurs whereas the total fields are continuous. Formonochromatic forward and backward propagating waves

a(t, z) = a(z)ejωt and b(t, z) = b(z)ejωt (2.215)

propagating in z-direction over a distance z with a propagation constant k,we find from Eqs.(2.202) and (2.203)

µa(z)b(z)

¶=

µe−jkz 00 ejkz

¶µa(0)b(0)

¶. (2.216)

A piece of transmission line can be described as a two port. The matrixtransforming the amplitudes of the waves at the input port (1) to those ofthe output port (2) is called the transfer matrix, see Figure 2.48

T

Figure 2.48: Definition of the wave amplitudes for the transfer matrix T.

For example, from Eq.(2.216) follows that the transfer matrix for freespace propagation is

µa2b2

¶= T

µa1b1

¶, with T =

µe−jkz 00 ejkz

¶. (2.217)

This formalism can be expanded to arbitrary multiports, see Figure 2.49.The scattering matrix describes then the transformation from incoming andoutgoing wave amplitudes of the multiport.


S

Figure 2.49: Scattering matrix and its port definition.

The scattering matrix defines a linear transformation from the incomingto the outgoing waves

b = Sa, with a = (a1,a2, ...)T , b = (b1,b2, ...)

T . (2.218)

Note, that the meaning between forward and backward waves no longer co-incides with a and b, a connection, which is difficult to maintain if severalports come in from many different directions.The transfer matrix T has advantages, if many two ports are connected

in series with each other. Then the total transfer matrix is the product ofthe individual transfer matrices.

2.4.4 Properties of the Scattering Matrix

Physial properties of the system reflect itself in the mathematical propertiesof the scattering matrix.

Reciprocity

A system with constant scalar dielectric and magnetic properties must havea symmetric scattering matrix (without proof)

S = ST . (2.219)


Losslessness

In a lossless system the total power flowing into the system must be equal tothe power flowing out of the system in steady state

|a|2 =¯b¯2. (2.220)

ThereforeS+S = 1 or S−1=S+. (2.221)

The scattering matrix of a lossless system must be unitary.

Time Reversal

To find the scattering matrix of the time reversed system, we realize thatincoming waves become outgoing waves under time reversal and the otherway around, i.e. the meaning of a and b is exchanged and on top of it thewaves become negative frequency waves.

aej(ωt−kz)time reversal→ aej(−ωt−kz). (2.222)

To obtain the complex amplitude of the corresponding positive frequencywave, we need to take the complex conjugate value. So to obtain the equa-tions for the time reversed system we have to perform the following substi-tutions

Original system Time reversed systemb = Sa a∗ = Sb

∗ → b =¡S−1

¢∗a. (2.223)

2.4.5 Beamsplitter

As an example, we look at the scattering matrix for a partially transmittingmirror, which could be simply formed by the interface between two mediawith different refractive index, which we analyzed in the previous section,see Figure 2.50. (Note, for brevity we neglect the reflections at the normalsurface input to the media, or we put an AR-coating on them.) In principle,this device has four ports and should be described by a 4x4 matrix. However,most often only one of the waves is used at each port, as shown in Figure2.50.


Figure 2.50: Port definitions for the beam splitter

The scattering matrix is determined by

b = Sa, with a = (a1,a2)T , b = (b1, b2)

T (2.224)

and

S=

µr jtjt r

¶, with r2 + t2 = 1. (2.225)

The matrix S was obtained using the S-matrix properties described above.From Eqs.(2.122) we could immediately identify r as a function of the re-fractive indices, angle of incidence and the polarization used. Note, that theoff-diagonal elements of S are identical, which is a consequence of reciprocity.That the main diagonal elements are identical is a consequence of unitarityfor a lossless beamsplitter and furthermore t =

√1− r2. For a given fre-

quency r and t can always be made real by choosing proper reference planesat the input and the output of the beam splitter. Beamsplitters can be madein many ways, see for example Figure 2.37.

2.4.6 Interferometers

Having a valid description of a beamsplitter at hand, we can build and ana-lyze various types of interferometers, see Figure 2.51.


Figure 2.51: Different types of interferometers: (a) Mach-Zehnder Interfer-ometer; (b) Michelson Interferometer; (c) Sagnac Interferometer [6] p. 66.

Each of these structures has advantages and disadvantages dependingon the technology they are realized. The interferometer in Figure 2.51 (a)is called Mach-Zehnder interferometer, the one in Figure 2.51 (b) is calledMichelson Interferometer. In the Sagnac interferometer , Figure 2.51 (c) bothbeams see identical beam path and therefore errors in the beam path can bebalance out and only differential changes due to external influences lead toan output signal, for example rotation, see problem set 3.To understand the light transmission through an interferometer we ana-

lyze as an example the Mach-Zehnder interferometer shown in Figure 2.52.If we excite input port 1 with a wave with complex amplitude a0 and noinput at port 2 and assume 50/50 beamsplitters, the first beam splitter will


φ2

φ1

Figure 2.52: Mach-Zehnder Interferometer

produce two waves with complex amplitudes

b1 =1√2a0

b2 = j 1√2a0

(2.226)

During propagation through the interferometer arms, both waves pick up aphase delay φ1 = kL1 and φ2 = kL2, respectively

a3 =1√2a0e

−jφ1 ,

a4 = j 1√2a0e

−jφ2 .(2.227)

After the second beam splitter with the same scattering matrix as the firstone, we obtain

b3 =12a0¡e−jφ1 − e−jφ2

¢,

b4 = j 12a0¡e−jφ1 + e−jφ2

¢.

(2.228)

The transmitted power to the output ports is

|b3|2 = |a0|2

4

¯1− e−j(φ1−φ2)

¯2= |a0|

2

2[1− cos (φ1 − φ2)] ,

|b4|2 = |a0|2

4

¯1 + e−j(φ1−φ2)

¯2= |a0|

2

2[1 + cos (φ1 − φ2)] .

(2.229)

The total output power is equal to the input power, as it must be for a losslesssystem. However, depending on the phase difference ∆φ = φ1 − φ2 betweenboth arms, the power is split differently between the two output ports, seeFigure 2.53.With proper biasing, i.e. φ1 − φ2 = π/2 + ∆φ0, the difference


1.0

0.8

0.6

0.4

0.2

0.0O

utpu

t pow

er-3 -2 -1 0 1 2 3

Δφ

|b3|2

|b4|2

Figure 2.53: Output power from the two arms of an interfereometer as afunction of phase difference.

in output power between the two arms can be made directly proportional tothe phase difference ∆φ0 for ∆φ0 ¿ 1.

Opening up the beam size in the interferometer and placing optics intothe beam enables to visualize beam distortions due to imperfect optical com-ponents, see Figures 2.54 and 2.55.

Figure 2.54: Twyman-Green Interferometer to test optics quality [1] p. 324.


Figure 2.55: Interference pattern with a hot iron placed in one arm of theinterferometer ([1], p. 395).

2.4.7 Multipath Interference and Fabry-Perot Resonator

Interferometers can act as filters. The phase difference between the interfer-ometer arms depends on frequency, therefore, the transmission from inputto output depends on frequency, see Figure 2.53. However, the filter functionis not very sharp. The reason for this is that only a two beam interferenceis used. Much more narrowband filters can be constructed by multipath in-terferences such as in a Fabry-Perot Resonator, see Figure 2.56. To indicateagain the these wave amplitudes are in the Fourier Domain, we added thetilde. The simplest Fabry Perot is described by a sequence of three layerswhere at least the middle layer has an index different from the other twolayers, such that reflections occur on these interfaces.


S

n2n1 n3

S

1

21

2

~

~

~ ~ ~

~~~

~~

~~

~~ ~

~

Figure 2.56: Multiple intereferences in a Fabry Perot resonator. In the sim-plest implementation a Fabry Perot only consists of a sequence of three layerswith different refractive index so that two reflections occur with multiple in-terferences. Each of this discontinuites can be described by a scatteringmatrix.

Any kind of device that has reflections at two parallel interfaces mayact as a Fabry Perot such as two semitransparent mirrors. A thin layerof material against air can act as a Fabry-Perot and is often called etalon.Given the reflection and transmission coefficients at the interfaces 1 and 2,we can write down the scattering matrices for both interfaces according toEqs.(2.224) and (2.225).µ

b1b2

¶=

µr1 jt1jt1 r1

¶µa1a2

¶and

µb3b4

¶=

µr2 jt2jt2 r2

¶µa3a4

¶.

(2.230)If we excite the Fabry-Perot with a wave from the right with amplitudea1 6= 0, then a fraction of that wave will be transmitted to the interface intothe Fabry-Perot as wave b2 and part will be already reflected into b1,

b(0)

1 = r1a1. (2.231)

The transmitted wave will then propagate and pick up a phase factor e−jφ/2,with φ = 2k2L and k2 =

2πλn2,

a3=jta1e−jφ/2. (2.232)


After propagation it will be reflected off from the second interface which hasa reflection coefficient

Γ2 =b3a3

¯¯a4=0

= r2. (2.233)

Then the reflected wave b3 propagates back to interface 1, picking up anotherphase factor e−jφ/2 resulting in an incoming wave after one roundtrip of a(1)2 =jt1r2e

−jφa1. Upon reflection on interface 1, part of this wave is transmittedleading to an output

b(1)

1 =jt1jt1r2e−jφa1. (2.234)

The partial wave a(1)2 is reflected again and after another roundtrip it arrivesat interface 1 as a(2)2 = (r1r2) e

−jφ · jt1r2e−jφa1. Part of this wave is trans-mitted and part of it is reflected back to go through another cycle. Thus intotal if we sum up all partial waves that contribute to the output at port 1of the Fabry-Perot filter, we obtain

b1 =∞Xn=0

b(n)

1

=

Ãr1 − t21r2e

−jφ∞Xn=0

r1r2e−jφ

!a1

=

µr1 − t21r2

e−jφ

1− r1r2e−jφ

¶a1

=r1 − r2e

−jφ

1− r1r2e−jφa1 (2.235)

Note, that the coefficient in front of Eq.(2.235) is the coefficient S11 of thescattering matrix of the Fabry-Perot. In a similar manner, we obtainµ

b1b4

¶= S

µa1a4

¶(2.236)

and

S=1

1− r1r2e−jφ

µr1 − r2e

−jφ −t1t2e−jφ/2−t1t2e−jφ/2 r2 − r1e

−jφ

¶(2.237)

In the following, we want to analyze the properties of the Fabry-Perot forthe case of symmetric reflectors, i.e. r1 = r2 and t1 = t2. Then we obtain for


the power transmission coefficient of the Fabry-Perot, |S21|2 in terms of thepower reflectivity of the interfaces R = r2

|S21|2 =¯1−R

1−Re−jφ

¯2=

(1−R)2

(1−R)2 + 4R sin2(φ/2)(2.238)

Figure 2.57 shows the transmission |S21|2 of the Fabry-Perot interferometerfor equal reflectivities |r1|2 = |r2|2 = R.

1.0

0.8

0.6

0.4

0.2

0.0Fabr

y-P

erot

Tra

nsm

issi

on, |

S 21|2

3.02.52.01.51.00.50.0(f -fm)/ FSR

R=0.1

R=0.5

R=0.7

R=0.9

FSR

ΔfFWHM

Figure 2.57: Transmission of a lossless Fabry-Perot interferometer with|r1|2 = |r2|2 = R

At very low reflectivity R of the mirror the transmission is almost every-where 1, there is only a slight sinusoidal modulation due to the first orderinterferences which are periodically in phase and out of phase, leading to100% transmission or small reflection. For large reflectivity R, due to thethen multiple interference operation of the Fabry-Perot Interferometer, verynarrow transmission resonances emerge at frequencies, where the roundtripphase in the resonator is equal to a multiple of 2π

φ =2πf

c0n22L = 2πm, (2.239)


which occurs at a comb of frequencies, see Figure 2.58

fm = mc02n2L

. (2.240)

1.0

0.8

0.6

0.4

0.2

0.0Fabr

y-P

erot

Tran

smis

sion

,|S

21|2

54321 fm

am

fmfm-1 fm+1 fm+2

am-1 am+1 am+2

frequency

Figure 2.58: Developement of a set of discrete resonances in a one-dimensionsal resonator.

On a large frequency scale, a set of discrete frequencies, resonances ormodes arise. The frequency range between resonances is called free spectralrange (FSR) of the Fabry-Perot Interferometer

FSR =c02n2L

=1

TR, (2.241)

which is the inverse roundtrip time TR of the light in the one-dimensonalcavity or resonator formed by the mirrors. The filter characteristic of eachresonance can be approximately described by a Lorentzian line derived fromEq.(2.238) by substituting f = fm +∆f with ∆f ¿ FSR,

|S21|2 =(1−R)2

(1−R)2 + 4R sin2¡£m2π + 2π ∆f

FSR

¤/2¢

≈ 1

1 +³2π√R

1−R∆fFSR

´2 , (2.242)

≈ 1

1 +³

∆f∆fFWHM/2

´2 , (2.243)


where we introduced the FWHM of the transmission filter

∆fFWHM =FSR

F, (2.244)

with the finesse of the interferometer defined as

F =π√R

1−R≈ π

T. (2.245)

The last simplification is valid for a highly reflecting mirror R ≈ 1 and T isthe mirror transmission. From this relation it is immediately clear that thefinesse has the additional physical meaning of the optical power enhancementinside the Fabry-Perot at resonance besides the factor of π, since the powerinside the cavity must be larger by 1/T , if the transmission through theFabry-Perot is unity.

2.4.8 Quality Factor of Fabry-Perot Resonances

Another quantity often used to characterize a resonator or a resonance isits quality factor Q, which is defined as the ratio between the resonancefrequency and the decay rate for the energy stored in the resonator, which isalso often called inverse photon lifetime, τ−1ph

Q = τ phfm. (2.246)

Lets assume, energy is stored in one of the resonator modes which occupies arange of frequencies [fm − FSR/2, fm + FSR/2] as indicated in Figure 2.59.Then the fourier integral

am(t) =

Z +FSR/2

−FSR/2b2(f)e

j2π(f−fm)t df, (2.247)

where¯b2(f)

¯2is normalized such that it describes the power spectral density

of the forward traveling wave in the resonator gives the mode amplitude of them-th mode and its magnitude square is the energy stored in the mode. Note,that we could have taken any of the internal waves a2, b2, a3, and b3. The timedependent field we create corresponds to the field of the forward or backwardtraveling wave at the corresponding reference plane in the resonator.


1.0

0.8

0.6

0.4

0.2

0.0Tran

smis

sion

, |S 2

1|2

3.02.52.01.51.00.50.0(f -fm)/ FSR

am(t) am+1(t)am-1(t)

Figure 2.59: Integration over all frequency components within the frequencyrange [fm − FSR/2, fm + FSR/2] defines a mode amplitude a(t) with a slowtime dependence

We now make a "Gedanken-Experiment". We switch on the incomingwaves a1(ω) and a4(ω) to load the cavity with energy and evaluate the in-ternal wave b2(ω). Instead of summing up all the multiple reflections likewe did in constructing the scattering matrix (2.236), we exploit our skillsin analyzing feedback systems, which the Fabry-Perot filter is. The scat-tering equations set force by the two scattering matrices characterizing theresonator mirrors in the Fabry-Perot can be visuallized by the signal flowdiagram in Figure 2.60

j t+

r

+

a2_~

a1_~b2_~

b1_~

j t

r

j t+

r

+

a4_~

a3_~ b4_~

b3_~

j t

r

e-jφ/2

e-jφ/2

S1 S2

Figure 2.60: Representation of Fabry-Perot resonator by a signal flow dia-gram


For the task to find the relationship between the internal waves feed bythe incoming wave only the dashed part of the signal flow is important. Theinternal feedback loop can be clearly recognized with a closed loop transferfunction

r2e−jφ,

which leads to the resonance denominator

1− r2e−jφ

in every element of the Fabry-Perot scattering matrix (2.236). Using Blacksformula from 6.003 and the superposition principle we immediately find forthe internal wave

b2 =jt

1− r2e−jφ¡a1 + re−jφ/2a4

¢. (2.248)

Close to one of the resonance frequencies, Ω = 2πfm + ω, using t = 1− r2,the round-trip phase can be approximated by

φ = ΩTR = 2πm+ ωTR, (2.249)

and the exponential of the round-trip phase by

e−jφ/2 = (−1)me−jωTR/2, (2.250)

e−jφ = e−jωTR = 1− jωTR. (2.251)

Using those approximations in (2.248) and t =√1−R we obtain

b2(ω) ≈j√1−R

1−R(1− jωTR)

¡a1(ω) + r(−1)me−jωTR/2a4(ω)

¢, (2.252)

≈ j

1 + j R1−RωTR

1√T

¡a1(ω) + r(−1)me−jωTR/2a4(ω)

¢(2.253)

≈ j

1 + jωTR/T

1√T

¡a1(ω) + r(−1)me−jωTR/2a4(ω)

¢(2.254)

for high reflectivity R. Multiplication of this equation with the resonant de-nominator

(1 + jωTR/T ) b2(ω) =j√T

¡a1(ω) + r(−1)me−jωTR/2a4(ω)

¢(2.255)


and inverse Fourier-Transform in the time domain, while recognizing thatthe internal fields vanish far off resonance, i.e.

am(t) =

Z +π·FSR

−π·FSRb2(ω)e

jωt dω =

Z +∞

−∞b2(ω)e

jωt dω, (2.256)

we obtain the following differential equation for the mode amplitude slowlyvarying in time

TRd

dtam(t) = −Tam(t) + j

√T (a1(t) + (−1)ma4(t− TR/2)) (2.257)

with the input fields

a1/4(t) =

Z +π·FSR

−π·FSRa1/4(ω)e

jωt dω. (2.258)

Despite the pain to derive this equation the physical interpretation is remark-ably simple and far reaching as we will see when we apply this equation laterto many different situations. Lets assume, we switch off the loading of thecavity at some point, i.e. a1/4(t) = 0, then Eq.(2.257) results in

am(t) = am(0)e−t/(TR/T ) (2.259)

And the power decays accordingly

|am(t)|2 = |am(0)|

2 e−t/(TR/2T ) (2.260)

twice as fast as the amplitude. The energy decay time of the cavity is oftencalled the cavity energy decay time, or photon lifetime, τ ph, which is here

τ ph =TR2T

.

Note, the factor of two comes from the fact that each mirror of the Fabry-Perot has a transmission T per round-trip time. Thus the total transmissionor loss form the cavity is 2T . For exampl a L = 1.5m long cavity with mirrorsof 0.5% transmission, i.e. TR = 10ns and 2T = 0.01 has a photon lifetime of1μs. It needs hundred bounces on the mirror for a photon to be essentiallylost from the cavity.Highest quality dielectric mirrors may have a reflection loss of only 10−5...−6,

this is not really transmission but rather scattering loss in the mirror. Such


high reflectivity mirrors may lead to the construction of cavities with photonlifetimes on the order of milliseconds.Now, that we have an expression for the energy decay time in the cavity,

we can evaluate the quality factor of the resonator

Q = fm · τ ph =m

2T. (2.261)

Again for a resonator with the same parameters as before and at opticalfrequencies of 300THz corresponding to 1μm wavelength, we obtain Q =2 · 108.

2.4.9 Thin-Film Filters

Transfer matrix formlism is an efficient method to analyze the reflection andtransmission properties of layered dielectric media, such as the one shownin Figure 2.61. Using the transfer matrix method, it is an easy task tocompute the transmission and reflection coefficients of a structure composedof layers with arbitrary indices and thicknesses. A prominent example of athin-film filter are Bragg mirrors. These are made of a periodic arrangementof two layers with low and high index n1 and n2, respectively. For maximumreflection bandwidth, the layer thicknesses are chosen to be quarter wave forthe wavelength maximum reflection occures, n1d1 =λ0/4 and n2d2 =λ0/4

a2_~

a1_~ b2_~

b1_~

....n1 n1 n1n2n2 n2

d1 d1 d1d2 d2 d2

Figure 2.61: Thin-Film dielectric mirror composed of alternating high andlow index layers.

As an example Figure 2.62 shows the reflection from a Bragg mirror withn1 = 1.45, n2 = 2.4 for a center wavelength of λ0 = 800nm. The layerthicknesses are then d1 =134nm and d2 =83nm.


500 600 700 800 900 1000 1100 1200 13000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

lambda

refle

ctiv

ity

Figure 2.62: Reflectivity of an 8 pair quarter wave Bragg mirror with n1 =1.45 and n2 = 2.4 designed for a center wavelength of 800nm. The mirror isembedded in the same low index material.


2.5 Gaussian Beams and Resonators

So far, we have only treated optical systems operating with plane waves,which is an idealization. In reality plane waves are impossible to generatebecause of there infinite amount of energy required to do so. The simplest(approximate) solution of Maxwell’s equations describing a beam of finite sizeis the Gaussian beam. In fact many optical systems are based on Gaussianbeams. Most lasers are designed to generate a Gaussian beam as output.Gaussian beams stay Gaussian beams when propagating in free space. How-ever, due to its finite size, diffraction changes the size of the beam and lensesare imployed to reimage and change the cross section of the beam. In thissection, we want to study the properties of Gaussian beams and how theycan be used to understand the spatial modes of optical resonators.

2.5.1 Paraxial Wave Equation

We start from the Helmholtz Equation (2.17)

¡∆+ k20

¢ eE(x, y, z, ω) = 0, (2.262)

with the free space wavenumber k0 = ω/c0. This equation can easily besolved in the Fourier domain, and one set of solutions are of course the planewaves with wave vector |k|2 = k20. We look for solutions which are polarizedin x-direction e

E(x, y, z, ω) = eE(x, y, z) ex. (2.263)

We want to construct a beam with finite transverse extent into the x-y-planeand which is mainly propagating into the positive z-direction. As such wemay try a superposition of plane waves with a dominant z-component of thek-vector, see Figure 2.63. The k-vectors can be written as

kz =qk20 − k2x − k2y,

≈ k0

µ1−

k2x + k2y2k20

¶. (2.264)

with kx, ky << k0.

2.5. GAUSSIAN BEAMS AND RESONATORS 103

z

k

yx

Figure 2.63: Construction of a paraxial beam by superimposing many planewaves with a dominante k-component in z-direction.

Then we obtain for the propagating field

eE(x, y, z) =

Z +∞

−∞

Z +∞

−∞eE0(kx, ky) ·

exp

∙−jk0

µ1−

k2x + k2y2k20

¶z − jkxx− jkyy

¸dkxdky,

=

Z +∞

−∞

Z +∞

−∞eE0(kx, ky) ·

exp

∙j

µk2x + k2y2k0


¸dkxdkye

−jk0z, (2.265)

where eE0(kx, ky) is the amplitude for the waves with the corresponding trans-verse k-component. This function should only be nonzero within a smallrange kx, ky ¿ k0. The function

eE0(x, y, z) = Z +∞

−∞

Z +∞

−∞eE0(kx, ky) exp ∙jµk2x + k2y

2k0


¸dkxdky

(2.266)is a slowly varying function in the transverse directions x and y, and it canbe easily verified that it fulfills the paraxial wave equation

∂

∂zeE0(x, y, z) = −j

2k0

µ∂2

∂x2+

∂2

∂x2

¶ eE0(x, y, z). (2.267)

Note, that this equation is in its structure identical to the dispersive spreadingof an optical pulse. The difference is that this spreading occurs now in thetwo transverse dimensions and is called diffraction.


2.5.2 Gaussian Beams

Since the kernel in Eq.(2.266) is quadratic in the transverse k-componentsusing a two-dimensional Gaussian for the amplitude distribution leads to abeam in real space which is also Gaussian in the radial direction because ofthe resulting Gaussian integral. By choosing for the transverse amplitudedistribution eE0(kx, ky) ∼ exp ∙−k2x + k2y

2k2T

¸, (2.268)

Eq.(2.266) can be rewritten as

eE0(x, y, z) ∼ Z +∞

−∞

Z +∞

−∞exp

∙j

µk2x + k2y2k0

¶(z + jzR)− jkxx− jkyy

¸dkxdky,

(2.269)with the parameter zR = k0/k

2T , which we will later identify as the Rayleigh

range. Thus, Gaussian beam solutions with different finite transverse widthin k-space and real space behave as if they propagate along the z-axis withdifferent imaginary z-component zR. Carrying out the Fourier transformationresults in the Gaussian Beam in real space

eE0(x, y, z) ∼ j

z + jzRexp

∙−jk0

µx2 + y2

2(z + jzR)

¶¸. (2.270)

The Gaussian beam is often formulated in terms of the complex beam pa-rameter or q-parameter.The propagation of the beam in free space and later even through optical

imaging systems can be efficiently described by a proper transformation ofthe q-parameter

eE0(r, z) ∼ 1

q(z)exp

∙−jk0

µr2

2q(z)

¶¸. (2.271)

Free space propagation is then described by

q(z) = z + jzR (2.272)

Using the inverse q-parameter, decomposed in real and imagniary parts,

1

q(z)=

1

R(z)− j

λ

πw2(z). (2.273)


leads to

eE0(r, z) = p4ZF0P√πw(z)

exp

∙− r2

w2(z)− jk0

r2

2R(z)+ jζ(z)

¸. (2.274)

Thus w(z) is the waist of the beam and R(z) is the radius of the phasefronts. We normalized the beam such that the Gaussian beam intensity

I(z, r) =¯ eE0(r, z)¯2 /(2ZF0) expressed in terms of the power P carried by the

beam is given by

I(r, z) =2P

πw2(z)exp

∙− 2r2

w2(z)

¸, (2.275)

i.e. P =

Z ∞

0

Z 2π

0

I(r, z) rdr dϕ. (2.276)

The use of the q-parameter simplifies the description of Gaussian beam prop-agation. In free space propagation from z1 to z2, the variation of the beamparameter q is simply governed by

q2 = q1 + z2 − z1. (2.277)

where q2 and q1 are the beam parameters at z1 and z2.If the beam waist, at which the beam has a minimum spot size w0 and

a planar wavefront (R = ∞), is located at z = 0, the variations of thebeam spot size and the radius of curvature of the phase fronts are explicitlyexpressed as

w(z) = wo

"1 +

µz

zR

¶2#1/2, (2.278)

and

R(z) = z

∙1 +

³zRz

´2¸, (2.279)

where zR is called the Rayleigh range. The Rayleigh range is the distanceover which the cross section of the beam doubles. The Rayleigh range isdetermined by the beam waist and the wavelength of light according to

zR =πw2oλ

. (2.280)


Intensity

Figure 2.64 shows the intensity of the Gaussian beam according to Eq.(2.275)for different propagation distances.

Figure 2.64: The normalized beam intensity I/I0 as a function of the radialdistance r at different axial distances: (a) z=0, (b) z=zR, (c) z=2zR.

The beam intensity can be rewritten as

I(r, z) = I0w20

w2(z)exp

∙− 2r2

w2(z)

¸, with I0 =

2P

πw20. (2.281)

For z > zR the beam radius growth linearly and therefore the area expandsquadratically, which brings down the peak intensity quadratically with prop-agation distance.On the beam axis (r = 0) the intensity is given by

I(r, z) = I0w20

w2(z)=

I0

1 +³

zzR

´2 . (2.282)

The normalized beam intensity as a function of propagation distance is shownin Figure 2.65


Figure 2.65: The normalized Beam intensity I(r = 0)/I0 on the beam axisas a function of propagation distance z [6], p. 84.

Power

The fraction of the total power contained in the beam up to a certain radiusis

P (r < r0)

P=

2π

P

Z r0

0

I(r, z)rdr

=4

w2(z)

Z r0

0

exp

∙− 2r2

w2(z)

¸rdr (2.283)

= 1− exp∙− 2r20w2(z)

¸.

Thus, there is a certain fraction of power within a certain radius of thebeam

P (r < w(z))

P= 0.86, (2.284)

P (r < 1.5w(z))

P= 0.99. (2.285)

Beam radius

Due to diffraction, the smaller the spot size at the beam waist, the faster thebeam diverges according to 2.278 as illustrated in Figure ??.


Planes of const. Phase

Beam Waist

z/z R

Figure 2.66: Gaussian beam and its characteristics.

Beam divergence

The angular divergence of the beam is inversely proportional to the beamwaist. In the far field, the half angle divergence is given by

θ =λ

πwo, (2.286)

see Figure 2.66.

Confocal parameter and depth of focus

In linear microscopy, only a layer which has the thickness over which thebeam is focused, called depth of focus, will contribute to a sharp image. Innonlinear microscopy (see problem set) only a volume on the order of beamcross section times depth of focus contributes to the signal. Therefore, thedepth of focus or confocal parameter of the Gaussian beam, is the distanceover which the beam stays focused and is defined as twice the Rayleigh range

b = 2zR =2πw2oλ

. (2.287)

The confocal parameter depends linear on the spot size (area) of the beamand is inverse to the wavelength of light. At a wavelength of 1μm a beamwith a radius of wo = 1cm,.the beam will stay focussed ove distances as long


600m. However, if the beam is stronlgy focussed down to wo = 10μm thefield of depth is only 600μm.

Phase

The phase delay of the Gaussian beam is

Φ(r, z) = k0z − ζ(z) + k0r2

2R(z)(2.288)

ζ(z) = arctan

µz

zR

¶. (2.289)

On beam axis, there is the additional phase ζ(z) when the beam undergoesfocussing as shown in Figure 2.67. This is in addition to the phase shift thata uniform plane wave already aquires.

Figure 2.67: Phase delay of a Gaussian beam relative to a uniform plane waveon the beam axis [6], p. 87. This phase shift is known as Guoy-Phase-Shift.

This effect is known as Guoy-Phase-Shift. The third term in the phaseshift is parabolic in the radius and describes the wavefront (planes of constantphase) bending due to the focusing, i.e. distortion from the uniform planewave.


Figure 2.68: The radius of curvature R(z) of the wavefronts of a Gaussianbeam [6], p. 89.

The surfaces of constant phase are determined by k0z − ζ(z) + k0r2

2R(z)=

const. Since the radius of curvature R(z) and the additional phase ζ(z) areslowly varying functions of z, i.e. they are constant over the radial variationof the wavefront, the wavefronts are paraboloidal surfaces with radius R(z),see Figures 2.68 and 2.69.

Figure 2.69: Wavefronts of a Gaussian beam, [6] p. 88.

For comparison, Figure 2.70 shows the wavefront of (a) a uniform planewave, (b) a spherical wave and (c) a Gaussian beam. At points near thebeam center, the Gaussian beam resembles a plane wave. At large z, thebeam behaves like a spherical wave except that the phase fronts are delayedby a quarter of the wavlength due to the Guoy-Phase-Shift.


Figure 2.70: Wavefronts of (a) a uniform plane wave;(b) a spherical wave;(c) a Gaussian beam [5], p. 89.

2.5.3 Optical Resonators

With the Gaussian beam solutions, we can finally construct optical resonatorswith finite transverse extent, i.e. real Fabry-Perots. To do that we insert intothe Gaussian beam, see Figure 2.71, curved mirrors with the proper radiusof curvature, such that the beam is imaged upon itself.


z1

L

z2

R1 R2

Figure 2.71: Fabry-Perot resonator with finite beam cross section by insertingcurved mirrors into the beam to back reflect the beam onto itself.

Curved-Flat Mirror Resonator

We first consider the simple resonator constructed by a curved and flat mirror,i.e. only the left side of the gaussian beam in Figure 2.71, see Figure 2.72

z1 L

R1

=

Figure 2.72: Curved-Flat Mirror Resonator

From Eqs.(2.278) and (2.279) we immediately get an expression for theradius of curvature of the mirror necessary to generate a certain confocalparameter zR =

πw2oλor spot size on the flat mirror given a certain wavelength

w1 = wo

"1 +

µL

zR

¶2#1/2, (2.290)


and

R1 = L

∙1 +

³zRL

´2¸. (2.291)

Since the confocal parameter is a positive quantity it follows that we alwaysneed a mirror with a radius of curvature R1 > L, to form a stable gaussianmode in the resonator. The spot size on the flat and curved mirrors in termsof the radius of curvature of the mirror and the distance of the mirror is

wo =

rλR1π

4

sL

R1

µ1− L

R1

¶, (2.292)

and

w1 =

rλR1π

4

vuut LR1

1− LR1

. (2.293)

There is a resonator mode with finite size for 0 < L < R1, i.e. the cavityis stable in this parameter range.. Figure 2.73 shows the normalized beamradii on both mirrors as a function of the normalized distance between themirrors L

R1.

3

2

1

0

Bea

m R

adii

1.00.80.60.40.20.0L/R1

w1/sqrt(R1λ/π)

w0/sqrt(R1λ/π)

Figure 2.73: Beam waists of the curved-flat mirror resonator as a function ofthe cavity parameter g1.


Note, that for a given curved mirror with radius of curvature (ROC) =R1, the waist of the beam goes to zero for both extremes of the distancebetween the mirrors where the cavity is stable. There is a large central rangearound L ≈ R1/2, where the beam waist is maximum and insensitive to theexact distance between the mirrors.

Two-Curved Mirror Resonator

Now, we can analyze what happens for the case of a resonator with two curvedmirrors as shown in Figure 2.71. The gaussian beam formulas Eqs.(2.278)and (2.279) give us now the following conditions for beam waists

w1 = wo

"1 +

µz1zR

¶2#1/2, (2.294)

w2 = wo

"1 +

µz2zR

¶2#1/2, (2.295)

R1 = z1

"1 +

µzRz1

¶2#, (2.296)

R2 = z2

"1 +

µzRz2

¶2#, (2.297)

andL = z1 + z2. (2.298)

The last three equations determine the confocal parameter zR and the dis-tances between location of waist and mirrors z1 and z2 in dependence on theradii of curvature R1, R2 and the distance between mirrors L. It is straightforward to eliminate the confocal parameter from Eqs.(2.296), (2.297) andto solve the remaining equation together with Eq.(2.298) for the distancesz1 and z2. The result is

z1 =L(R2 − L)

R1 +R2 − 2L, (2.299)

and by symmetry

z2 =L(R1 − L)

R1 +R2 − 2L= L− z1. (2.300)


Resubstitution into Eq.(2.296) leads to an expression for the Rayleigh rangeof the beam and its waist

z2R = L(R1 − L)(R2 − L)(R1 +R2 − L)

(R1 +R2 − 2L)2. (2.301)

and therefore

w4o =

µλL

π

¶2(R1 − L)(R2 − L)(R1 +R2 − L)

L(R1 +R2 − 2L)2(2.302)

=

µλ√R1R2π

¶2L2

R1R2

(1− LR1)(1− L

R2)( L

R1+ L

R2− L

R1LR2)

( LR1+ L

R2−2 L

R1LR2)2

. (2.303)

For completeness the spot size on the mirrors is given by

w41 =

µλR1π

¶2R2 − L

R1 − L

µL

R1 +R2 − L

¶(2.304)

=

µλ√R1R2π

¶2 1− LR2

1− LR1

LR1

LR2

LR1+ L

R2− L

R1LR2

. (2.305)

By symmetry, we find the spot size on mirror 2 by switching index 1 and 2:

w42 =

µλR2π

¶2R1 − L

R2 − L

µL

R1 +R2 − L

¶(2.306)

=

µλ√R1R2π

¶2 1− LR1

1− LR2

LR1

LR2

LR1+ L

R2− L

R1LR2

. (2.307)

The mode characteristics w0, w1, w2, z1 and z2for the two-mirror resonatorare shown in Figure 2.74 for the caseR1 = 10cm and R2 = 11cm as a functionof the mirror distance L


0.6

0.4

0.2

0.0w0 /

(λ (

R1*

R2)

1/2/π

)1/2

20151050Cavity Length, L / cm6

4

2

0w1 /

(λ R

1/π )

1/2

20151050Cavity Length, L / cm

6

4

2

0

w2 /

(λ R

2/π )

1/2

20151050Cavity Length, L / cm1.0

0.80.60.40.20.0

z 1 / L


1.00.80.60.40.20.0

z2 / L

-1.0-0.50.00.51.0

g 1, g

2


1.00.80.60.40.20.0

S = g1 x g

2

Figure 2.74: From top to bottom: two-mirror resonator mode characteristicsw0, w1, w2, z1, z2 and cavity parameters, g1, g2, S for the case R1 = 10 cmand R2 = 11 cm.


Resonator stability As we can read of from Figure 2.74 for a two-mirrorresonator with concave mirrors and R1 ≤ R2, we obtain the general stabilitydiagram as shown in Figure 2.75.

Figure 2.75: Stabile regions (black) for the two-mirror resonator.

There are two ranges for the mirror distance L, within which the cavityis stable, 0 ≤ L ≤ R1 and R2 ≤ L ≤ R1 + R2.The stable and unstableparameter ranges can be expressed in a compact way in terms of the cavityparameters g1 and g2 defined by gi = (Ri − L)/Ri, for i = 1, 2.These cavityparameters together and its product S = g1 · g2 are shown in the last graphof Figure 2.74. The cavity mode has finite size, i.e. it is stable for

stable : 0 ≤ g1 · g2 = S ≤ 1 (2.308)

and unstable for

unstable : g1g2 ≤ 0; or g1g2 ≥ 1. (2.309)

The stability criterion can be easily interpreted geometrically. Of importanceare the distances between the mirror mid-points Mi and the cavity end points,i.e. gi = (Ri − L)/Ri = −Si/Ri, as shown in Figure 2.76.


Figure 2.76: The stability criterion involves distances between the mirrormid-points Mi and the cavity end points. i.e. gi = (Ri − L)/Ri = −Si/Ri.

The following rules for a stable resonator can be derived from Figure 2.76using the stability criterion expressed in terms of the distances Si. Note, thatwithout proof, the distances and radii can be positive and negative, wherethe latter indicate concave mirrors

stable : 0 ≤ S1S2R1R2

≤ 1. (2.310)

This results in the following rules for the geometry of stable two-mirror res-onators:

• A resonator is stable if the mirror radii, laid out along the optical axis,overlap.

• A resonator is unstable if the radii do not overlap or one lies within theother.

Figure 2.77 shows stable and unstable resonator configurations.


Figure 2.77: Illustration of stable and unstable resonator configurations.

Hermite-Gaussian-Beams (TEMmn-Beams)

It turns out that the Gaussian Beams are not the only solution to the parax-ial wave equation (2.267). The stable modes of the resonator reproducethemselves after one round-trip,

eEm,n(x, y, z) = Am,n

∙w0w(z)

¸Gm

"√2x

w(z)

#Gn

∙√2y

w(z)

¸· (2.311)

exp

∙−jk0

µx2 + y2

2R(z)

¶+ j(m+ n+ 1)ζ(z)

¸

where

Gm [u] = Hm [u] exp

∙−u

2

2

¸, for m = 0, 1, 2, ... (2.312)


are the Hermite-Gaussians with the Hermite-Polynomials

H0 [u] = 1,

H1 [u] = 2u,

H2 [u] = 4u2 − 1, (2.313)

H3 [u] = 8u3 − 12u,

and ζ(z) is the Guoy-Phase-Shift according to Eq.(2.289). The lower orderHermite Gaussians are depicted in Figure 2.78

Figure 2.78: Hermite-Gauissians Gm(u) for m = 0, 1, 2 and 3.

and the intensity profile of the first higher order resonator modes areshown in Figure 2.79.


Figure 2.79: Intensity profile of TEMmn-beams, [6], p. 103.

Besides the different mode profiles, the higher order modes experiencegreater phase advances during propogation, because they are made up ofk-vectors with larger transverse components.

Axial Mode Structure

As we have seen for the Fabry-Perot resonator, the longitudinal modes arecharacterized by a roundtrip phase that is a multiple of 2π. Back then, wedid not consider transverse modes. Thus in a resonator with finite transversebeam size, we obtain an extended family of resonances, with distinguish-able field patterns. The resonance frequencies ωpmn are determined by theroundtrip phase condition

φplm = 2pπ, for p = 0,±1,±2, ... (2.314)


For the linear resonator according to Figure 2.71, the roundtrip phase of aHermite-Gaussian Tplm-beam is

φplm = 2kL− 2(l +m+ 1) (ζ(z2)− ζ(z1)) , (2.315)

where ζ(z2)− ζ(z1) is the additional Guoy-Phase-Shift, when the beam goesthrough the focus once on its way from mirror 1 to mirror 2. Then theresonance circular frequencies are

ωplm =c

L[πp+ (l +m+ 1) (ζ(z2)− ζ(z1))] (2.316)

If the Guoy-Phase-Shift is not a rational number times π, then all resonancefrequencies are non degenerate. However, for the special case where thetwo mirrors have identical radius of curvature R and are spaced a distanceL = R apart, which is called a confocal resonator, the Guoy-Phase-shift isζ(z2)− ζ(z1) = π/2, with resonance frequencies

fplm =c

2L

∙p+

1

2(l +m+ 1)

¸. (2.317)

In that case all even, i.e. l + m, transverse modes are degenerate to thelongitudinal or fundamental modes, see Figure 2.80.

Figure 2.80: Resonance frequencies of the confocal Fabry-Perot resonator,[6], p. 128.

The odd modes are half way inbetween the longitudinal modes. Note, incontrast to the plan parallel Fabry Perot all mode frequencies are shifted byhalf of the free spectral range, FSR = c

2L, due to the Guoy-Phase-Shift.

2.6. WAVEGUIDES AND INTEGRATED OPTICS 123

2.6 Waveguides and Integrated Optics

As with electronics, miniaturization and integration of optics is desired toreduce cost while increasing functionality and reliability. One essential el-ement is the guiding of the optical radiation in waveguides for integratedoptical devices and optical fibers for long distance transmission. Waveguidescan be as short as a few millimeters. Guiding of light with exceptionally lowloss in fiber (0.1dB/km) can be achieved by using total internal reflection.Figure 2.81 shows different optical waveguides with a high index core mate-rial and low index cladding. The light will be guided in the high index core.Similar to the Gaussian beam the guided mode is made up of mostly paraxialplane waves that hit the high/low-index interface at grazing incidence andtherefore undergo total internal reflections. The concomittant lensing effectovercomes the diffraction of the beam that would happen in free space andleads to stationary mode profiles for the radiation.Depending on the index profile and geometry one distinguishes between

different waveguide types. Figure 2.81 (a) is a planar slab waveguide, whichguides light only in one direction. This case is analyzed in more detail,as it has simple analytical solutions that show all phenomena associatedwith waveguiding such as cutoff, dispersion, single and multimode operation,coupling of modes and more, which are used later in devices and to achievecertain device properties. The other two cases show complete waveguidingin the transverse direction; (b) planar strip waveguide and (c) optical fiber.

Saleh 239

Figure 2.81: Dark shaded area constitute the high index regions. (a) planarslab waveguide; (b) strip waveguide; (c) optical fiber [6], p. 239.

In integrated optics many components are fabricated on a single sub-


strate, see Figure 2.82 with fabrication processes similar to those in micro-electronics.

(a)

(b)

Figure 2.82: (a) Schematic of an integrated polarization insensitive OpticalAdd-Drop Multiplexer (OADM). It integrates polarization splitters, rotators,and ring filter stages for each polarization to drop and add a correspondingwavelength channel for each polarization. The two polarziations are laterrecombined to form a single output into a fiber. (b) The device utilizes SiNwaveguides on silica using a dual mask lithography process. [11].

More advanced devices also contain modulators, lasers and photo detec-tors to perform complete transmitt and receive functions. But as the examplein Fig. 2.82 shows, the most important passive component to understand inan integrated optical circuit are waveguides and couplers. Here, we treat thebasics of these devices.

2.6.1 Planar Waveguides

To understand the basic physics and phenomena in waveguides, we look ata few examples of guiding in one transverse dimension. These simple casescan be treated analytically.


Planar-Mirror Waveguides

The planar mirror waveguide is composed of two ideal metal mirrors a dis-tance d apart, see Figure 2.83

yx

Figure 2.83: Planar mirror waveguide, [6], p. 240.

We consider a TE-wave, whose electric field is polarized in the y−directionand that propagates in the z−direction. The reflections of the light at theideal lossless mirrors will guide or confine the light in the x−direction. Thefield will be homogenous in the y−direction, i.e. will not depend on y. There-fore, we make the following trial solution for the electric field of a monochro-matic complex TE-wave

E(x, z, t) = Ey(x, z) ejωt ey. (2.318)

Note, this trial solution also satisfies the condition ∇ ·E = 0, see (2.6)

Modes of the planar waveguide Furthermore, we are looking for solu-tions that do not change their field distribution transverse to the directionof propagation and experience only a phase shift during propagation. Wecall such solutions modes of the waveguide, because they don’t change itstransverse field profile. The modes of the above planar waveguide can beexpressed as

Ey(x, z) = u(x) e−jβz, (2.319)

where β is the propagation constant of the mode. This solution has to obeythe Helmholtz Eq.(2.17) in the free space section between the mirrors

d2

dx2u(x) =

¡β2 − k2

¢u(x) with k2 =

ω2

c2. (2.320)


The presence of the metal mirrors requires that the electric field vanishes atthe metal mirrors, otherwise infinitely strong currents would start to flow toshorten the electric field.

u(x = ±d/2) = 0 (2.321)

.Note, that Eq.(2.320) is an eigenvalue problem to the differential operatord2

dx2

d2

dx2u(x) = λu(x) with u(x = ±d/2) = 0. (2.322)

in a space of functions u, that satisfies the boundary conditions (2.321). Theeigenvalues λ are for the moment arbitrary but constant numbers. Dependingon the sign of the eigenvalues the solutions can be sine or cosine functions(λ < 0) or exponentials with real exponents for (λ > 0). In the latter case, it isimpossible to satiesfy the boundary conditions. Therefore, the eigensolutionsare

um(x) =

⎧⎨⎩q

2dcos (kx,mx) with , kx,m =

πdm, m = 1, 3, 5, ..., even modesq

2dsin (kx,mx) with , kx,m =

πdm, m = 2, 4, 6, ..., odd modes

(2.323)

Propagation Constants The propagation constants for these modes fol-low from comparing (2.320) with (2.322) to be

β2m = k2 − k2x,m (2.324)

or

βm = ±r

ω2

c2−³πdm´2= ±

sµ2π

λ

¶2−³πdm´2

(2.325)

where λ = λ0/n(λ0) is the wavelength in the medium between the mirrors.This relationship is shown in Figure 2.84. The lowest order mode with indexm = 1 has the smallest k-vector component in x-direction and therefore thelargest k-vector component into z-direction. The sum of the squares of bothcomponents has to be identical to the magnitude sqaure of the k-vector inthe medium k. Higher order modes have increasingly more nodes in thex-direction, i.e. largest kx-components and the wave vector component in z-direction decreases, until there is no real solution anymore to Eq.(2.325) andthe corresponding propagation constants βm become imaginary. That is, the


corresponding waves become evanescent waves, i..e they can not propagatein a waveguide with the given dimensions.

β

0

k

πd

πd

πd

πd1 2 3 4

β

ββ

k

12

3

kx

Figure 2.84: Determination of propagation constants for modes

Field Distribution The transverse electric field distributions for the var-ious TE-modes is shown in Figure 2.85

yx

Figure 2.85: Field distributions of the TE-modes of the planar mirror waveg-uide [6], p. 244.

CutoffWavelength/Frequency For a given planar waveguide with sep-aration d, there is a lowest frequency, i.e. longest wavelength, beyond whichno propagating mode exists. This wavelenth/frequency is refered to as cutoff


wavelength/frequency which is

λcutoff = 2d (2.326)

fcutoff =c

2d. (2.327)

The physical origin for the existence of a cutoff wavelength or frequency isthat the guided modes in the mirror waveguide are a superposition of twoplane waves, that propagate under a certain angle towards the z-axis, seeFigure 2.86

x

Figure 2.86: (a) Condition for self-consistency: as a wave reflects twice itneeds to be in phase with the previous wave. (b) The angles for which self-consistency is achieved determine the x-component of the k-vectors involved.The corresponding two plane waves setup an interference pattern with anextended node at the position of the metal mirrors satisfying the boundaryconditions, [6], p. 241.

In order that the sum of the electric field of the two plane waves fulfillsthe boundary conditions, the phase of one of the plane waves after reflection


on both mirrors needs to be inphase with the other plane wave, i.e. thex-component of the k-vectors involved, kx, must be a multiple of 2π

2kxd = ±2πm.

If we superimpose two plane waves with kx,m = ±πm/d, we obtain an in-terference pattern which has nodes along the location of the metal mirrors,which obviously fulfills the boundary conditions. It is clear that the mini-mum distance between these lines of nodes for waves of a given wavelength λis λ/2, hence the separation dmust be greater than λ/2 otherwise no solutionis possible.

Single-Mode Operation For a given separation d, there is a wavelengthrange over which only a single mode can propagate, we call this wavelengthrange single-mode operation. From Figure 2.84 it follows for the planarmirror waveguide

π

d< k <

π

d2 (2.328)

ord < λ < 2d (2.329)

Waveguide Dispersion Due to the waveguiding, the relationship betweenfrequency and propagation constant is no longer linear. This does not implythat the waveguide core, i.e. here the medium between the plan parallelmirrors, has dispersion. For example, even for n = 1, we find for phase andgroup velocity of the m-th mode

1

vp,m=

βm(ω)

ω=1

c

r1−

³ cπdω

m´2

(2.330)

=1

c

s1−

µλ

2dm

¶2(2.331)

and1

vg,m=

dβm(ω)

dω=

1

2q

ω2

c2−¡πdm¢22ωc2 (2.332)

orvg,m · vp,m = c2. (2.333)


Thus different modes have different group and phase velocities. Figure 2.87shows group and phase velocity for the different modes as a function of thenormalized wave number kd/π.

2.0

1.5

1.0

0.5

0.0

v g/c

an

d v p

/c

543210kd/π

m=1

m=1

m=2m=3

m=4

m=2 m=3m=4

vg

vp

Figure 2.87: Group and phase velocity of propagating modes with index mas a function of normailzed wave number.

TM-Modes The planar mirror waveguide does not only allow for TE-waves to propagate. There are also TM-waves, which have only a magneticfield component transverse to the propagation direction and parallel to themirrors, i.e. in y-direction

H(x, z, t) = Hy(x, z) ejωt ey, (2.334)

and now H(x, z) has to obey the Helmholtz equation for the magnetic field.The corresponding electric field can be derived from Ampere’s law

E(x, z) =1

jωε∇×

¡Hy(x, z) ey

¢(2.335)

= − 1

jωε

∂Hy(x, z)

∂zex +

1

jωε

∂Hy(x, z)

∂xez. (2.336)

The electric field tangential to the metal mirrors has to vanish again, whichleads to the boundary condition

∂Hy(x, z)

∂x(x = ±d/2) = 0. (2.337)


After an analysis very similar to the discussion of the TE-waves we find forthe TM-modes with

Hy(x, z) = u(x) e−jβz ey, (2.338)

the transverse mode shapes

um(x) =

⎧⎨⎩q

2dcos (kx,mx) with , kx,m =

πdm, m = 0, 2, 4, 6, ..., even modesq

2dsin (kx,mx) with , kx,m =

πdm, m = 1, 3, 5, ..., odd modes

(2.339)Note, that in contrast to the electric field of the TE-waves, which is zeroat the metal surface, the transverse magnetic field of theTM-waves is at amaximum at the metal surface. Also there is no cutoff wavelenght of theTM-modes. When the frequency goes to zero, the TM-wave becomes a TEMwave between the two conducting plates. We will not consider this casefurther, because otherwise the discussion of modes and waveguide dispersioncan be worked out very analogous to the case for TE-modes.

Multimode Propagation Depending on the boundary conditions at theinput of the waveguide at z = 0 many modes may be excited. Eventuallythere are even excitations with such high transverse wavevectors kx present,that are below cutoff. Depending on the excitation amplitudes of each mode,the total field in the waveguide will be the superposition of all modes. Letsassume that there are only TE-modes excited, then the total field is

E(x, z, t) =∞X

m=1

¡am e−jβmz + bm ejβmz

¢um(x) e

jωt ey, (2.340)

where the amplitudes am and bm are the excitations of the m-th mode inforward and backward direction, respectively. It is easy to show that theseexcitation amplitudes are determined by the transverse electric and magneticfields at z = 0 and t = 0. In many cases, the excitation of the waveguide willbe such that only the forward propagating modes are excited.

E(x, z, t) =∞X

m=1

am um(x) e−jβmz ejωt ey, (2.341)

When many modes are excited, the transverse field distribution will changeduring propagation, see Figure 2.88


x

x

x

Figure 2.88: Variation of the intensity distribution in the transverse directionx at different distances z. Intensity profile of (a) the fundamental modem = 1, (b) the second mode with m = 2 and (c) a linear combination of thefundamental and second mode, [6], p. 247.

Modes which are excited below cutoff will decay rapidly as evanescentwaves. The other modes will propagate, but due to the different propaga-tion constants these modes superimpose differently at different propagationdistances along the waveguide. This dynamic can be used to build manykinds of important integrated optical devices, such as multimode interfer-ence couplers (see problem set 5). Depending on the application, undesiredmultimode excitation may be very disturbing due to the large group delaydifference between the different modes. This effect is called modal dispersion.

Mode Orthogonality

It turns out that the transverse modes determined by the functions um(x)build an orthogonal set of basis functions into which any function in a cer-tain function space can be decomposed. This is obvious for the case of the


planar-mirror waveguide, where the um(x) are a subset of the basis functionsfor a Fourier series expansion of an arbitrary function f(x) in the interval[−d/2, d/2] which is fullfills the boundary condition f(x = ±d/2) = 0. It is

Z d/2

−d/2um(x) un(x) dx = δmn, (2.342)

f(x) =Xm

am um(x) (2.343)

with am =

Z d/2

−d/2um(x) f(x) dx (2.344)

From our familiarity with Fourier series expansions of periodic functions,we can accept these relations here without proof. We will return to theseequations later in Quantum Mechanics and discuss in which mathematicalsense Eqs.(2.342) to (2.343) really hold.

Besides illustrating many important concepts, the planar mirror waveg-uide is not of much practical use at optical frequencies. More in use aredielectric waveguides.

Planar Dielectric Slab Waveguide

In the planar dielectric slab waveguide, waveguiding is not achieved by realreflection on a mirror but rather by total internal reflection at interfacesbetween two dielectric materials with refractive indices n1 > n2, see Figure2.89


x

y

Figure 2.89: Symmetric planar dielectric slab waveguide with n1 > n2. Thelight is guided by total internal reflection. The field is evanescent in thecladding material and oscillatory in the core, [6], p. 249.

Waveguide Modes As in the case of the planar mirror waveguide, thereare TE and TM-modes and we could find them as a superposition of cor-respondingly polarized TEM waves propagating with a certain transversek-vector such that total internal reflection occurs. We do not want to followthis procedure here, but rather use immediately the Helmholtz Equation. Weagain write the electric field

Ey(x, z) = u(x) e−jβz ey. (2.345)

The field has to obey the Helmholtz Eq.(2.17) both in the core and in thecladding

core :d2

dx2u(x) =

¡β2 − k21

¢u(x) with k21 =

ω2

c20n21, (2.346)

cladding :d2

dx2u(x) =

¡β2 − k22

¢u(x) with k22 =

ω2

c20n22 (2.347)

The boundary conditions are given by the continuity of electric and magneticfield components tangential to the core/cladding interfaces as in section 2.2.


Since the guided fields must be evanescent in the cladding and oscillatory inthe core, we rewrite the Helmholtz Equation as

core :d2

dx2u(x) = −k2xu(x) with k2x =

¡k21 − β2

¢, (2.348)

cladding :d2

dx2u(x) = κ2xu(x) with κ2x =

¡β2 − k22

¢(2.349)

where κx is the decay constant of the evanescent waves in the cladding. Itis obvious that for obtaining guided modes, the propagation constant of themode must be between the two propagation constants for core and cladding

k22 < β2 < k21. (2.350)

Or by defining an effective index for the mode

β = k0neff , with k0 =ω

c0(2.351)

we find

n1 > neff > n2, (2.352)

and Eqs.(2.348), (2.349) can be rewritten as

core : − d2

dx2u(x)− k20

¡n21 − n2eff

¢u(x) = 0 (2.353)

cladding : − d2

dx2u(x)− k20

¡n22 − n2eff

¢u(x) = 0 (2.354)

For reasons, which will become more obvious later, we draw in Figure 2.90the negative refractive index profile of the waveguide.


-d/2 d/2 x

-neff

-n1

-n2

0

Figure 2.90: Negative refractive index profile and shape of electric field forthe fundamental and first higher order transverse TE-mode

From Eq.(2.353) we find that the solution has the general form

u(x) =

⎧⎨⎩ A exp (−κxx) + B exp (κxx) , for x < −d/2C cos (kxx) + D sin (kxx) , for |x| < d/2E exp (−κxx) + F exp (κxx) , for x > d/2

(2.355)

For a guided wave, i.e. um(x → ±∞) = 0 the coefficients A and F mustbe zero. It can be also shown from the symmetry of the problem, that thesolutons are either even or odd (proof later)

u(e)(x) =

⎧⎨⎩ B exp (κxx) , for x < −d/2C cos (kxx) , for |x| < d/2E exp (−κxx) , for x > d/2

, (2.356)

u(o)(x) =

⎧⎨⎩ B exp (κxx) , for x < −d/2D sin (kxx) , for |x| < d/2E exp (−κxx) , for x > d/2

. (2.357)

The coefficients B and E in each case have to be determined from the bound-ary conditions. From the continuity of the tangential electric field Ey, and


the tangential magnetic field Hz, which follows from Faraday’s Law to be

Hz(x) =1

−jωμ0∂Ey

∂x∼ du

dx(2.358)

we obtain the boundary conditions for u(x)

u(x = ±d/2 + ) = u(x = ±d/2− ), (2.359)du

dx(x = ±d/2 + ) =

du

dx(x = ±d/2− ). (2.360)

Note, these are four conditions determining the coefficients B,D,E and thepropagation constant β or refractive index neff . These conditions solve forthe parameters of even and odd modes separately. For the case of the evenmodes, where B = E, we obtain

B exp

µ−κx

d

2

¶= C cos

µkxd

2

¶(2.361)

B κx exp

µ−κx

d

2

¶= Ckx sin

µkxd

2

¶(2.362)

or by division of both equations

κx = kx tan

µkxd

2

¶. (2.363)

Eqs.(2.348) and (2.349) can be rewritten as one equation

k2x + κ2x =¡k21 − k22

¢= k20

¡n21 − n22

¢(2.364)

Eq.(2.363) together with Eq.(2.364) determine the propagation constant βvia the two relations.

κxd

2= kx

d

2tan

µkxd

2

¶, and (2.365)µ

kxd

2

¶2+

µκx

d

2

¶2=

µk0d

2NA

¶2(2.366)

whereNA =

q(n21 − n22) (2.367)


is called the numerical apperture of the waveguide. We will discuss thephysical significance of the numerical apperture shortly. A graphical solutionof these two equations can be found by showing both relations in one plot,see Figure 2.91.

10

8

6

4

2

0

κ xd/2

1086420kxd/2

m=0m=1

m=2m=3

m=4 m=5

k0d/2 NA

Figure 2.91: Graphical solution of Eqs.(2.365) and ( 2.366), solid line foreven modes and Eq.(2.368) for the odd modes. The dash dotted line shows(2.366) for different values of the product

¡k0

d2

¢NA

Each crossing in Figure 2.91 of a solid line (2.365) with a circle (2.366)with radius k0 d2NA represents an even guided mode. Similarly one finds forthe odd modes from the boundary conditions the relation

κxd

2= −kx

d

2cot

µkxd

2

¶, (2.368)

which is shown in Figure 2.91 as dotted line. The corresponding crossingswith the circle indicate the existence of an odd mode.There are also TM-modes, which we don’t want to discuss for the sake of

brevity.

Numerical Aperture Figure 2.91 shows that the number of modes guidedis determined by he product k0 d2NA, where NA is the numerical apperture


defined in Eq.(2.367)

M = Int

∙k0d

2NA/(π/2)

¸+ 1, (2.369)

= Int

∙2d

λ0NA

¸+ 1, (2.370)

where the function Int[x] means the largest integer not greater than x. Note,that there is always at least one guided mode no matter how small the sizedand the refractive index contrast between core and cladding of the waveguideis. However, for small size and index contrast the mode may extend very farinto the cladding and the confinement in the core is low.The numerical apperture also has an additional physical meaning that

becomes obvious from Figure 2.92.

Figure 2.92: Maximum angle of incoming wave guided by a waveguide withnumerical aperture NA, [6], p. 262.

The maximum angle of an incoming ray that can still be guided in thewaveguide is given by the numerical apperture, because according to Snell’sLaw

n0sin (θa) = n1sin (θ) , (2.371)

where n0 is the refractive index of the medium outside the waveguide. Themaximum internal angle θ where light is still guided in the waveguide bytotal internal reflection is determined by the critical angle for total internalreflection (2.135) , i.e. θmax = π/2− θtot with

sin (θtot) =n2n1

. (2.372)


Thus for the maximum angle of an incoming ray that can still be guided wefind

n0sin (θa,max) = n1sin (θmax) = n1

s1−

µn2n1

¶2= NA. (2.373)

Most often the external medium is air with n0 ≈ 1 and the refractive indexcontrast is week, so that θa,max ¿ 1 and we can replace the sinusoid with itsargument, which leads to

θa,max = NA. (2.374)

Field Distributions Figure 2.93 shows the field distribution for the TEguided modes in a dielectric waveguide. Note, these are solutions of thesecond order differential equations (2.353) and (2.354) for an effective indexneff , that is between the core and cladding index. These guided modes havea oscillatory behavior in those regions in space where the negative effectiveindex is larger than the negative local refractive index, see Figure 2.90 andexponentially decaying solutions where the negative effective index is smallerthan than the negative local refractive index.

x

Figure 2.93: Field distributions for TE guided modes in a dielectric waveg-uide. These results should be compared with those shown in Figure 2.85 forthe planar-mirror waveguide [6], p. 254.

Figure 2.94 shows a comparison of the guided modes in a waveguidewith a Gaussian beam. In contrast to a the Gaussian beam which diffracts,


in a waveguide diffraction is balanced by the guiding action of the indexdiscontinuity, i.e. total internal reflection. Most importantly the cross sectionof a waveguide mode stays constant and therefore a waveguide mode canefficiently interact with the medium constituting the core or a medium thatis incorporated in the core.

Figure 2.94: Comparison of Gaussian beam in free space and a waveguidemode, [6], p. 255.

Besides integration, this prolonged interaction distance is one of the majorreasons for using waveguides. The interaction lenght can be arbitrarily long,only limited by the waveguide loss, in contrast to a Gaussian beam, whichstays focused only over the confocal distance or Rayleigh range.As in the case of a planar-mirror waveguide, one can show that the trans-

verse mode functions are orthogonal to each other. At first, a striking dif-ference here is that we have only a finite number of guided modes and onemight worry about the completeness of the transverse mode functions. Theanswer is that in addition to the guided modes, there are unguided modesor leaky modes, which together with the guided modes from a complete set.Each initial field can be decomposed into these modes. The leaky modesrapidly loose energy because of radiation and after a relatively short propa-gation distance only the field of guided modes remains in the waveguide. Wewill not pursue this further in this introductory class. The interested readershould consult with [12].


Confinement Factor

A very important quantity for a waveguide mode is its confinement in thecore, which is called the confinement factor

Γm =

R d/20

u2m(x) dxR∞0

u2m(x) dx. (2.375)

The confinement factor quantifies the fraction of the mode energy propagat-ing in the core of the waveguide. This is very important for the interactionof the mode with the medium of the core, which may be used to amplify themode or which may contain nonlinear media for frequency conversion.

Waveguide Dispersion

For the guided modes the effective refractive indices of the modes and there-fore the dispersion relations must be between the indices or dispersion rela-tions of core and cladding, see Figure 2.95

Figure 2.95: Dispersion relations for the different guided TE-Modes in thedielectric slab waveguide.

The different slopes dω/dβ for each mode indicate the difference in groupvelocity between the modes. Note, that there is at least always one guidedmode.


2.6.2 Two-Dimensional Waveguides

Both the planar-mirror waveguide and the planar dielectric slab waveguideconfine light only in one direction. It is straight forward to analyze the modesof the two-dimensional planar-mirror waveguide, which you have already donein 6.013. Figure 2.96 shows various waveguides that are used in praxis forvarious devices. Here, we do not want to analyze them any further, becausethis is only possible by numerical techniques.

Figure 2.96: Various types of waveguide geometries: (a) strip: (b) embeddedstrip: (c) rib ro ridge: (d) strip loaded. The darker the shading, the higherthe refractive index [6], p. 261.

2.6.3 Waveguide Coupling

The core size of a waveguide can range from a fraction of the free spacewavelength to many wavelength for a multimode fiber. For example a typicalhigh-index contrast waveguide with a silicon core and a silica cladding for1550 nm has a cross section of 0.2μm × 0.4μm, single-mode fiber, which wewill discuss in the next section with an index contrast of 0.5-1% between coreand cladding has a typical mode-field radius of 6μm.

If the mode cross section is not prohibitively small the simplest approachto couple light into a waveguide is by using a proper lens, see Figure 2.97(a) or direct butt coupling of the source to the waveguide if the source is awaveguide based device itself.


Figure 2.97: Coupling to a waveguide by (a) a lens; (b) direct butt couplingof an LED or laser diode, [6], p. 262

The lens and the beam size in free space must be chosen such that thespot size matches the size of the waveguide mode while the focusing angle infree space is less than the numerical aperture of the waveguide, (see problemset). Other alternatives are coupling to the evanscent field by using a prismcoupler, see Figure 2.98

Figure 2.98: Prism coupler, [6], p. 263.


Figure 2.99: Grating Coupler

The coupling with the prism coupler is maximum if the propagation con-stant of the waveguide mode matches the longitudinal component of thek-vector

β = knp cos θp,

Another way to match the longitudinal component of the k-vector of theincoming light to the propagation constant of the waveguide mode is by agrating coupler, see Figure 2.99

2.6.4 Coupling of Modes

If two dielectric waveguides are placed closely together their fields overlap.This situation is shown in Figure 2.100 at the example of the planar dielectricslab waveguide. Of course this situation can be achieved with any type oftwo dimensional dielectric waveguide shown in Figure 2.96


x

Figure 2.100: Coupling between the two modes of the dielectric slab waveg-uide, [6], p. 264.

Once the fields significantly overlap the two modes interact. The shapeof each mode does not change very much by the interaction. Therefore, wecan analyze this situation using perturbation theory. We assume that inzero-th order the mode in each waveguide is independent from the presenceof the other waveguide. We consider only the fundamental TE-modes ineach of the waveguide which have excitation amplitudes a1(z) and a2(z),respectively. The dynamics of each mode can be understood in terms of thiswave amplitude. In the absence of the second waveguide, each waveguideamplitude undergoes only a phase shift during propagation according to itsdispersion relations

da1(z)

dz= −jβ1a1(z), (2.376)

da2(z)

dz= −jβ2a2(z). (2.377)

The polarization generated by the field of mode 2 in waveguide 1 acts as asource for the field in waveguide 1 and the other way arround. Therefore,coupling of modes can be described by adding a source term in each equationproportional to the free propagation of the corresponding wave in the other


guide

da1(z)

dz= −jβ1a1(z)− jκ12a2(z), (2.378)

da2(z)

dz= −jκ21a1(z)− jβ2a2(z). (2.379)

κ12 and κ21 are the coupling constants of the modes. An expression in termsof waveguide properties can be derived using perturbation theory [6],. Thesecoupled mode equations describe a wealth of phenomena and are of funda-mental importance in many areas of physics and engineering.As we will see, there is only a significant interaction of the two modes if

the two propagation constants are not much different from each other (phasematching). Therefore, we write the propagation constants in terms of theaverage β0 and the phase mismatch ∆β

β1/2 = β0 ±∆β with (2.380)

β0 =β1 + β22

and ∆β =β1 − β22

. (2.381)

and we take the overall trivial phase shift of both modes out by introducingthe slowly varying relative field amplitudes

a1(z) = a1(z)ejβ0z and a2(z) = a2(z)e

jβ0z (2.382)

which obey the equation

d

dza1(z) = −j∆βa1(z)− jκ12a2(z), (2.383)

d

dza2(z) = −jκ21a1(z) + j∆βa2(z). (2.384)

Power conservation during propagation demands

d

dz

¡|a1(z)|

2 + |a2(z)|2¢ = 0 (2.385)

which requests that κ21 = κ∗12, i.e. the two coupling coefficients are notindependent from each other (see problem set).Note, Eqs.(2.383) and (2.384) are a system of two linear ordinary differ-

ential equations with constant coefficients, which is straight forward to solve


(see problem set). Given the excitation amplitudes a1(0) and a2(0) = 0 atthe input of the waveguides, i.e. no input in waveguide 2 the solution is

a1(z) = a1(0)

µcos γz − j

∆β

γsin γz

¶, (2.386)

a2(z) = −ja1(0)κ21γsin γz, (2.387)

with

γ =

q∆β2 + |κ12|2. (2.388)

The optical powers after a propagation distance z in both waveguides arethen

P1(z) = |a1(z)|2 = P1(0)

Ãcos2 γz +

µ∆β

γ

¶2sin2 γz

!, (2.389)

P2(z) = P1(0)

Ã|κ21|2

γ2

!sin2 γz. (2.390)

This solution shows, that depending on the difference in phase velocity be-tween the two-waveguides more or less power is coupled back and fourthbetween the two waveguides, see Figure 2.101.The period at which the power exchange occurs is

L =π

γ. (2.391)

If both waveguides are identical, i.e. ∆β = 0 and γ = |κ12|, the waves arephase matched, Eqs.(2.389) and (2.390) simplify to

P1(z) = P1(0) cos2 γz (2.392)

P2(z) = P1(0) sin2 γz. (2.393)

Complete transfer of power occurs between the two waveguides after a dis-tance

L0 =π

2 |κ12|, (2.394)

see Figure 2.102


Figure 2.101: Periodic exchange of power between guides 1 and 2 [6], p. 266.

Figure 2.102: Exchange of power between guides 1 and 2 in the phase-matched case, [6], p. 266.

Depending on the length of the coupling region the coupling ratio can bechosen. A device with a distance L0/2 and L0 achieves 50% and 100% powertransfer into waveguide two, respectively, see Figure 2.103


Figure 2.103: Optical couplers: (a) 100% coupler, (b) 3dB coupler, [6], 267.

2.6.5 Switching by Control of Phase Mismatch

If we keep the interaction length of the waveguides fixed at a length L0 =π

2|κ12| , then the power tranfer from waveguide 1 to waveguide 2 depends crit-ically on the phase mismatch ∆β

T (∆β) =P2(z = L0)

P1(z = 0)=³π2

´2sin c2

⎛⎝12

s1 +

µ2∆βL0

π

¶2⎞⎠ , (2.395)

where sinc(x) = sin(πx)/(πx). Figure 2.104 shows the transfer characteristicas a function of normalized phase mismatch. The phase mismatch betweenwaveguides can be controlled for example by the linear electro-optic or Pock-els effect, which we will investigate later.


Figure 2.104: Dependence of power transfer from waveguide 1 to waveguide2 as a function of phase mismatch, [6], p. 267.

The implementation of such a waveguide coupler switch is shown in Figure2.105.

Figure 2.105: Integrated waveguide coupler switch, [6], p. 708


2.6.6 Optical Fibers

Optical fibers are cylindrical waveguides, see Figure 2.106, made of low-lossmaterials such as silica glass.

Figure 2.106: Optical fibers are cylindrical dielectric waveguides, [6], p. 273.

Similar to the waveguides studied in the last section the most basic fibersconsist of a high index core and a lower index cladding. Today fiber technol-ogy is a highly developed art which has pushed many of the physical param-eters of a waveguide to values which have been thought to be impossible afew decades ago:

• Fiber with less than 0.16dB/km loss

• Photonic crystal fiber (Nanostructured fiber)

• Hollow core fiber

• Highly nonlinear fiber

• Er-doped fiber for amplifiers

• Yb-doped fiber for efficient lasers and amplifiers

• Raman gain fiber

• Large area single mode fibers for high power (kW) lasers.

Figure 2.107 shows the ranges of attenuation coefficients of silica glasssingle-mode and multimode fiber.


Figure 2.107: Ranges of attenuation coefficients of silica glass single-modeand multimode fiber, [10], p. 298.

For the purpose of this introductory class we only give an overview aboutthe mode structure of the most basic fiber, the step index fiber, see Figure2.108 (b)

Figure 2.108: Geometry, refractive index profile, and typical rays in: (a) amultimode step-index fiber, (b) a single-mode step-index fiber, (c) a multi-mode graded-index fiber [6], p. 274


Step-index fiber is a cylindrical dielectric waveguide specified by its coreand cladding refractive indices, n1 and n2 and the core radius a, see Fig-ure 2.106. Typically the cladding is assumed to be so thick that the finitecladding radius does not need to be taken into account. The guided modesneed to be sufficiently decayed before reaching the cladding boundary, whichis usually strongly scattering or absorbing. In standard fiber, the claddingindices differ only slightly, so that the relative refractive-index difference

∆ =n1 − n2

n1(2.396)

is small, typically 10−3 < ∆ < 2 ·10−2. Most fibers currently used in mediumto long optical communication systems are made of fused silica glass (SiO2) ofhigh chemical purity. The increase in refractive index of the core is achievedby doping with titanium, germanium or boron, among others. The refractiveindex n1 ranges from 1.44 to 1.46 depending on the wavelength utilized inthe fiber. The acceptance angle of the rays coupling from free space intoguided modes of the waveguide is determined by the numerical apperture asalready discussed for the dielectric slab waveguide, see Figure 2.109

θa ∼ sin(θa) = NA =qn21 − n22 ≈ n1

√2∆. (2.397)

Figure 2.109: The acceptance angle of a fiber and numerical aperture NA[6], p. 276.


Guided Waves

Again the guided waves can be found by looking at solutions of the Helmholtzequations in the core and cladding where the index is homogenous and byadditionally requesting the continuity of the tangential electric and magneticfields at the core-cladding boundary. In general the fiber modes are not any-longer pure TE or TM modes but rather are hybrid modes, i.e. the modeshave both transverse and longitudinal electric andmagnetic field components.Only the radial symmetric modes are still TE or TM modes. To determinethe exact mode solutions of the fiber is beyond the scope of this class andthe interested reader may consult reference [2]. However, for weakly guidingfibers, i.e. ∆¿ 1, the modes are actually very much TEM like, i.e. the longi-tudinal field components are much smaller than the radial field components.The linear in x and y directions polarized modes form orthogonal polariza-tion states. The linearly polarized (l,m) mode is usually denoted as theLPlm-mode.The two polarizations of the mode with indices (l,m) travel withthe same propagation constant and have identical intensity distributions.The generic solutions to the Helmhotz equation in cylindrical coordinates

are the ordinary, Jm(kr), and modified, Km(kr), Bessel functions (analogousto the cos(x)/ sin(x) and exponential functions e±κx, that are solutions to theHelmholtz equation in cartesian coordinates). Thus, a generic mode functionfor a cylinder symmetric fiber has the form

ul,m(r, ϕ) =

⎧⎪⎪⎨⎪⎪⎩Jl(kl,mr)

½cos(lϕ)sin(lϕ)

, for r < a, core

Kl(kl,mr)

½cos(lϕ)sin(lϕ)

, for r > a, cladding(2.398)

For large r, the modified Bessel function approaches an exponential,Kl(kl,mr) ∼

e−κlnmr

½cos(lϕ)sin(lϕ)

.The propagation constants for this two dimensional waveg-

uide have to fullfil the additional constraints

k2l,m =¡n21k

20 − β2

¢, (2.399)

κ2l,m =¡β2 − n22k

20

¢, (2.400)

k2l,m + κ2l,m = k20NA2. (2.401)

Figure 2.110 shows the radial dependence of the mode functions


Figure 2.110: Radial dependence of mode functions u(r),[6], p.279.

The transverse intensity distribution of the linearly polarized LP0,1 andLP3,4 modes are shown in Figure 2.111.

Figure 2.111: Intensity distribtuion of the (a) LP01 and (b) LP3,4 modes inthe transverse plane. The LP01 has a intensity distribution similar to theGaussian beam, [6], p. 283.

Number of Modes

It turns out, that as in the case of the dielectric slab waveguide the number ofguided modes critically depends on the numerical aperture or more precisely


on the V-parameter, see Eq.(2.369)

V = k0a

2NA. (2.402)

Without proof the number of modes is

M ≈ 4

π2V 2, for V À 1. (2.403)

which is similar to Eq.(2.369) for the one-dimensional dielectric slab waveg-uide, but the number of modes here is now related to the square of theV-parameter, because of the two-dimensional transverse confinement of themodes in the fiber. As in the case of the dielectric waveguide, there is al-ways at least one guided mode (two polarizations). However, the smaller theV-parameter the more the mode extends into the cladding and the guidingproperties become weak, i.e. small bending of the fiber may already lead tohigh loss.

Bibliography

[1] Hecht and Zajac, "Optics," Addison and Wesley, Publishing Co., 1979.

[2] B.E.A. Saleh and M.C. Teich, "Fundamentals of Photonics," JohnWileyand Sons, Inc., 1991.

[3] Bergmann and Schaefer, "Lehrbuch der Experimentalphysik: Opitk,"1993.

[4] H. Kogelnik and T. Li, ”Laser Beams and Resonators,” Appl. Opt. 5,pp. 1550 — 1566 (1966).

[5] H. Kogelnik, E. P. Ippen, A. Dienes and C. V. Shank, ”AstigmaticallyCompensated Cavities for CWDye Lasers,” IEEE J. Quantum Electron.QE-8, pp. 373 — 379 (1972).

[6] H. A. Haus, ”Fields and Waves in Optoelectronics”, Prentice Hall 1984.

[7] F. K. Kneubühl and M. W. Sigrist, ”Laser,” 3rd Edition, Teubner Ver-lag, Stuttgart (1991).

[8] A. E. Siegman, ”Lasers,” University Science Books, Mill Valley, Califor-nia (1986).

[9] Optical Electronics, A. Yariv, Holt, Rinehart & Winston, New York,1991.

[10] Photonic Devices, Jia, Ming-Liu , Cambridge University Press, 2005.

[11] T. Barwicz, et al., Polarization Transparent Microphotonics in theStrong Confinement Limit, in Nature Photonics 2007.

[12] T. Tamir, "Guided-Wave Optoelectronics," Springer, 1990.

159

Chapter 3

Quantum Nature of Light andMatter

We understand classical mechanical motion of particles governed by New-ton’s law. In the last chapter we examined in some detail the wave natureof electromagnetic fields. We understand the occurance of guided travelingmodes and of resonator modes. There are characteristic dispersion relationsor resonance frequencies associated with them. In this chapter, we want tosummarize some experimental findings that ultimately lead to the discoveryof quantum mechanics and with it to a greatly improved understanding ofthe nature of matter and electromagnetic waves. It turns out that matterhas in addition to its particle like properties which we are familiar with fromclasssical mechanics also wave properties and electromagnetic waves have inaddition to its wave properties particle like properties. The final theory,which will be developed in subsequent chapters is much more than just thatbecause the quantum mechanical wave function has a different physical inter-pretation than a electromagnetic wave only the mathematical concepts usedare in many cases very similar. However, familiarity with electromagneticwaves is a tremendous help and guideance in doing and finally understand-ing quantum mechanics.

3.1 Black Body Radiation

In 1900 the physicist Max Planck found the law that governs the emission ofelectromagnetic radiation from a black body in thermal equilibrium. More

161

162 CHAPTER 3. QUANTUM NATURE OF LIGHT AND MATTER

specifically Planck’s law gives the energy stored in the electromagntic field ina unit volume and unit frequency range, [f, f +df ] with df = 1Hz, when theelectromagnetic field is in thermal equilibrium with its surrounding that isat temperature T. A black body is simply defined as an object that absorbsall light. The best implementation of a black body is the Ulbricht sphere,see Figure 3.1.

Figure 3.1: The Ulbricht sphere, is a sphere with a small opening, whereonly a small amount of radiation can escape, so that the interior of thesphere is in thermal equilibrum with the walls, which are kept at a constanttremperature. The inside walls are typically made of diffuse material, so thatafter multiple scattering of the walls any incoming ray is absorbed, i.e. thewall opening is black.

Figure 3.2 shows the energy density w(f) of electromagnetic radiation ina black body at temperature T . Around the turn of the 19th century w(f)was measured with high precision and one was able to distinguish betweenvarious approximations that were presented by other researchers earlier, likethe Rayleigh-Jeans law and Wien’s law, which turned out to be asymptoticapproximations to Planck’s Law for low and high frequencies.In order to find the formula describing the graphs shown in Figure 3.2

Planck had to introduce the hypothesis that harmonic oscillators with fre-quency f can not exchange arbitrary amounts of energy but rather onlyin discrete portions, so called quanta. Planck modelled atoms as classical

3.1. BLACK BODY RADIATION 163

oscillators with frequency f . Therefore, the energy of an oscillator must bequantized in energy levels corresponding to these energy quanta, which hefound to be equal to hf, where h is Planck’s constant

h = 6.62620± 5 · 10−34Js. (3.1)

1.0

0.8

0.6

0.4

0.2

0.0

Bla

ck B

ody

Ene

rgy

Den

sity

w(f)

, fJs

/m3

1086420Frequency f, 100THz

Rayleigh-Jeans (T=5000K)

T=2000K

T=3000K

T=4000KT=5000K

Wien (T=5000K)

Figure 3.2: Spectral energy density of the black body radiation according toPlanck’s Law.

As a model for a black body, we neglect the presence of the opening, whichis only necessary to observe the radiation and use a cavity with perfectlyreflecting walls, somewhat different from the Ulbricht sphere. One couldmake this opening so small that it does essentially not change the internalradiation field. Then the radiation in the cavity is the sum over all possibleresonator modes in the cavity. If the cavity is at temperature T all themodes are thermally excited by emission and absorption of energy quantafrom the atoms of the wall.


For the derivation of Planck’s law we consider a cavity with perfectlyconducting walls, see Figure 3.3.

Figure 3.3: (a) Cavity resonator with metallic walls. (b) Resonator modescharacterized by a certain k-vector.

If we extend the analysis of the plan parallel mirror waveguide to find theTE and TM modes of a three-dimensional metalic resonator, the resonatormodes are TEmnp− and THmnp−modes characterized by its wave vector com-ponents in x−, y−, and z−direction. The resonances are standing waves inthree dimensions with wavevector components

kx =mπ

Lx, ky =

nπ

Ly, kz =

pπ

Lz, for m,n, p = 0, 1, 2, ... (3.2)

An expression for the number of modes in a frequency interval [f, f + df ]can be found by recognizing that this is identical to the number of points inFigure 3.3(b) that are in the first octant of a spherical shell with thicknessdk at k = 2πf/c.The volume occupied by one mode in the space of wavenumbers k is ∆V = π

Lx· πLy· πLz= π3

Vwith the volume V = LxLyLz. Then the

number of modes N(f) in the frequency interval [f, f + df ] in volume V are

N(f) = 2 · 4πk2dk

8π3

V

= Vk2dk

π2, (3.3)

where the factor of 2 in front accounts for the two polarizations or TE andTH-modes of the resonator and the 8 in the denominator accounts for the


fact that only 1/8th of the sphere, an octand, is occupied by the positivewave vectors. With k = 2πf/c and dk = 2πdf/c, we obtain for the numberof modes finally

N(f) = V8π

c3f2df (3.4)

Note, that the same density of states is obtained using periodic boundaryconditions in all three dimensions, i.e. then we can represent all fields interms of a three dimensional Fourier series. The possible wave vectors wouldrange from negative to positive values

kx =2mπ

Lx, ky =

2nπ

Ly, kz =

2pπ

Lzf or m, n, p = 0, ±1, ±2. . . . (3.5)

However, these wavevectors fill the whole sphere and not just 1/8th, whichcompensates for the 8-times larger volume occupied by one mode. If we implyperiodic boundary conditions, we have forward and backward running wavesthat are independent from each other. If we use the boundary conditionsof a resonator with perfectly conducting walls as before, the forward andbackward running waves are identical and not independent and form standingwaves. One should not be disturbed by this fact as all volume properties,such as the energy density, only depends on the density of states, and not onsurface effects, as long as the volume is reasonably large.

3.1.1 Rayleigh-Jeans-Law

The complex excitation amplitude a(t) of each mode obeys the equationof motion of a harmonic oscillator with a resonance frequency equal to theresonance frequency ω0 of the resonator mode.

a(t) = a(0)e−jω0t = Q(t) + jP (t) (3.6)

with

a(t) = −jω0 a(t) (3.7)

or

Q(t) + ω20 Q(t) = 0 (3.8)

Q(t) = ω0 P (t). (3.9)


Therefore, classically one expects that each mode is in thermal equilib-rium excited with a thermal energy kT according to the equipartition theo-rem, where k is Boltzmann’s constant with

k = 1.38062± 6 · 10−23J/K. (3.10)

If that is the case the spectral energy density is given by the Rayleigh-Jeans-Law, see Figure 3.2.

w(f) =1

V

N(f)

dfkT =

8π

c3f2kT. (3.11)

As can be seen from Figure 3.2, this law describes very well the black bodyradiation for frequencies hf ¿ kT but there is an arbitrary large deviationfor high frequencies. This formula can not be correct, because it predictsinfinite energy density for the high frequency modes resulting in an "ultravi-olet catastrophy", i.e. the electromagnetic field contains an infinite amountof energy at thermal equilibrium.

3.1.2 Wien’s Law

The high frequency or short wavelength region of the black body radiationwas first empirically described by Wien’s Law

w(f) =8πhf3

c3e−hf/kT . (3.12)

Wien’s law is surprisingly close to Planck’s law, however it slightly fails tocorrectly predict the asympthotic behaviour at low frequencies or long wave-lengths.

3.1.3 Planck’s Law

In the winter of 1900, Max Planck found the correct law for the black bodyradiation by assuming that each oscillator can only exchange energy in dis-crete portions or quanta. We rederive it by assuming that each mode canonly have the discrete energie values

Es = s · hf, for s = 0, 1, 2, ... (3.13)


Thus s is the number of energy quanta stored in the oscillator. If the oscillatoris a mode of the electromagnetic field we call s the number of photons. For theprobability ps, that the oscillator has the energy Es we assume a Boltzmann-distribution

ps =1

Zexp

µ−Es

kT

¶=1

Zexp

µ−hf

kTs

¶, (3.14)

where Z is a normalization factor such that the total propability of the os-cillator to have any of the allowed energy values is

∞Xs=0

ps = 1. (3.15)

Note, due to the fact that the oscillator energy is proportional to the numberof photons, the statistics are exponential statistics. From Eqs.(3.14) and(3.15) we obtain for the normalization factor

Z =∞Xs=0

exp

µ−hf

kTs

¶=

1

1− exp¡− hf

kT

¢ , (3.16)

which is also called the partition function. The photon statistics are thengiven by

ps = exp

µ−Es

kT

¶ ∙1− exp

µ−hf

kT

¶¸(3.17)

or with β = hfkT

ps =1

Z(β)e−βs , with Z (β) =

∞Xs=0

e−βs =1

1− e−β. (3.18)

Given the statistics of the photon number, we can compute moments of theprobability distribution, such as the average number of photons in the mode

hsi =∞Xs=0

s ps. (3.19)

This first moment of the photon statistics can be computed from the partitionfunction, using the "trick"

s1®=

1

Z(β)

∂1

∂ (−β)1Z(β) = Z(β) e−β , (3.20)


which is

hsi = 1

exp hfkT− 1

. (3.21)

With the average photon number hsi , we obtain for the average energy storedin the mode

hEsi = hsihf, (3.22)

and the energy density in the frequency intervall [f, f + df ] is then given by

w (f) =N(f)

V dfhEsi . (3.23)

With the density of modes from Eq.(3.4) we find Planck’s law for the blackbody radiation

w (f) =8π f 2

c3hf

exp hfkT− 1

, (3.24)

which was used to make the plots shown in Figure 3.2. In the limits of lowand high frequencies, i.e. hf ¿ kT and hf À kT , respectively Planck’s lawasympthotically approaches the Rayleigh-Jeans law and Wien’s law.

3.1.4 Thermal Photon Statistics

It is interesting to further investigate the intensity fluctuations of the thermalradiation emitted from a black body. If the wall opening in the Ulbrichtsphere, see Figure 3.1, is small enough very little radiation escapes throughit. If the Ulbricht sphere is kept at constant temperature the radiation insidethe Ulbricht sphere stays in thermal equilibrium and the intensity of theradiation emitted from the wall opening in a frequency interval [f, f + df ] is

I(f) = c · w (f) . (3.25)

Thus the intensity fluctuations of the emitted black body radiation is directlyrelated to the photon statistics or quantum statistics of the radiation modesat freuqency f , i.e. related to the stochastic variable s : the number ofphotons in a mode with frequency f . This gives us direclty experimentalaccess to the photon statistics of an ensemble of modes or even a single modewhen proper spatial and spectral filtering is applied.


Using the expectation value of the photon number 3.21, we can rewritethe photon statistics for a thermally excited mode in terms of its averagephoton number in the mode as

ps =hsis

(hsi+ 1)s+1=

1

(hsi+ 1)

µhsi

(hsi+ 1)

¶s

, (3.26)

The thermal photon statistics display an exponential distribution, see Figure3.4. Before we move on, lets see how the average photon number in a givenmode depends on temperature and the frequency range considered. Figure3.5 shows the relationship between average number of photons in a modewith frequency f or wavelength λ and temperature T.

Figure 3.4: Photon statistics of a mode in thermal equilibrium with a meanphoton number < s >= 10 (a) and < s >= 1000 (b).


Figure 3.5: Average photon number in a mode at frequency f or wavelengthλ and temperature.

Figure 3.5 shows that at room temperature and microwave frequencieslarge numbers of photons are present due to the thermal excitation of themode. However, at high frequencies, which start for room temperature in thefar to mid infrared range, on average much less than one photon is thermallyexcited.The variance of the photon number distribution is

σ2s =s2®− hsi2 . (3.27)

By generalizing Eq.(3.16) to the m-th moment by replacing the exponent 1by m

hsmi =∞Xs=0

smps (3.28)

=1

Z

∂m

∂ (−β)mZ (β) , (3.29)

we obtain for the second moments2®= 2Z (β) 2e−2β + Z (β) e−β = 2 hsi 2 + hsi . (3.30)

and therefore for the variance of the photon number using Eq.(3.27) is

σ2s = hsi2 + hsi . (3.31)


As expected from the wide distribution of photon numbers the variance islarger than the square of the expectation value. This means that if we lookat the light intensity coming form a single mode in thermal equilibrium, theintensity is subject to extremly strong fluctuations, i.e. its standard devi-ation, as large as its mean value. So why don’t we see this rapid thermalfluctuations when we look at the black body radiation coming, for example,from the surface of the sun? Well we don’t look at a single mode but ratherat a whole multitude of modes. Even when we restrict us to a certain nar-row frequency range and spatial direction, there is a multitude of transversemodes presence. We obtain for the average total number of photons in agroup of modes and its variance

hstoti =NXi=1

hsii , (3.32)

σ2tot =NXi=1

σ2i . (3.33)

Since these modes are independent identical systems, we have

hstoti = N · hsi . (3.34)

σ2tot = N¡hsi2 + hsi

¢=1

Nhstoti2 + hstoti .

Due to the averaging over many modes, the photon number fluctuations ina large number of modes is reduced compared to its mean value

SNR =hstoti2

σ2tot=

11N+ 1

hstoti. (3.35)

Thus, if one averages over many modes and has many photons in these modesthe intensity fluctuations become small. This is the reason while natural light,emitted from the sun or an incandescent lamp, both are thermal emitters,do not appear as highly fluctuating sources. Our eyes are averaging overmultiple modes which leads to a well defined average photon number hstotidetected in a certain time interval with a standard deviation σtot˜

phstoti

much smaller than the average vale for N →∞.


3.2 Photo-electric Effect

Another strong indication for the quantum nature of light was the discoveryof the photoelectric effect by Lenard in 1903. He discovered that when ultraviolet light is radiated on a photo cathode electrons are emitted, see Figure3.6.

-V

ffg

(a) (b)

Figure 3.6: Photo-electric effect: (a) Schematic setup and (b) dependence ofthe necessary grid voltage to supress the electron current as a funtion of lightfrequency.

Lenard surrounded the photo cathode by a grid, which is charged by theemitted electrons up to a voltage V, which blocks the emission of furtherelectrons. Figure 3.6 shows the blocking voltage as a function of the fre-quency of the incoming light. Depending on the cathode material there is acutoff frequency. For lower frequencies no electrons are emitted at all. Thisfrequency as well as the blocking voltage does not depend on the intensityof the light. In 1905, this effect was explained by Einstein introducing thequantum hypothesis for radiation. According to him, each electron emissionis caused by a light quantum, now called photon. This photon has an energyEph = hf and this quantum energy must be larger than the work functionWe of the material. The remainder of the energy mev

2/2 is transfered to theelectron in form of kinetic energy. The resulting energy balance is

hf =We +1

2mev

2 (3.36)

The kinetic energy of the electron can be used to reach the grid surrounding

3.3. PHOTODETECTION AND SHOT NOISE 173

the photo cathode until the charging energy due to the grid potential is equalto the kinetic energy of the electrons

−eV = 1

2mev

2 (3.37)

or

−V = 1

e(hf −We), for hf > We. (3.38)

This equation explains the empirically found law by Lenard explaining thecutoff frequency and the charge buildup as a function of light frequency.Einstein was first to introduce the idea that the electromagnetic field containslight quanta or photons.

3.3 Photodetection and Shot Noise

If a resistor is placed between the grid and the cathode, the voltage will drivean external current through the resistor and the arrangement is a photode-tector that converts the energy of the incoming photons into electrical energydissipated in a load resistor, see Fig. 3.7(a).

-V

I

(a) (b)

Figure 3.7: (a) Photodetector with external circuit. (b) Equivalent circuit.

The current delivered from the photodetector is

I = ηQE ·e

hfP = ηOEEP. (3.39)


where ηQE is called the quantum efficiency of the detector, e is the electroncharge, P the incoming optical power hitting the detector and ηOEE theresponsivity or optoelectronic efficiency (OEE) of the photo detector. Foran ideal photo detector ηQE = 1. The photon energy Eph depends on thewavelength according to

Eph = hf =1.24eV

λ[μm]. (3.40)

Thus the detector has at a wavelength of λ = 1.24μm a responsivity of

ηOEE = ηQE ·e

hf= 1A/W, (3.41)

i..e the detector delivers 1 Ampere electrical current for 1 Watt of opticalpower. Thus to first order a photodetector can be modelled as a currentsource, that delivers a current proportional to the optical power absorbed inthe detector, see Fig. 3.7(b).Without proof, it turns out that for an ideal photodector the statistics of

the electrons making up the photo current is identical to the statistic of theindicident flux of photons. In real photodetectors this one to one relation-ship between photons and electrons maybe spoiled by additional statisticalprocesses, like loss in the photodetector either on the optical or electrical sideresulting in ηQE < 1 and due to subsequent amplification of the electricalsignal. The statistical fluctuations in the photon number that arrive in agiven time interval carry over into statistical flucuations of the photo elec-trons making up the photo current. If the photo current I is made up of sphotons detected in a given time interval T, then this photo current can bedecomposed into its average current I and a fluctuating or often called noisecomponent In. The total current will be the average photo current and anoise photo current

I(t) = I + In(t). (3.42)

For an ideal photodetector the average current times the observation intervalT is identical to the average number of photons detected in that time interval

I

eT = hsi = number of detected photons. (3.43)

The noise current, which is a random variable is characterized by its powerspectral density SInIn(f). Each detector has a finite response time, i.e. the

3.3. PHOTODETECTION AND SHOT NOISE 175

time it takes for a photo electron to leave the photo detector, or in otherwords the photo current is made up of a sum of individual electrons emittedfrom the detector at random times ti

I(t) =Xi

e h(t− ti) with

∞Z−∞

h(t− ti) dt = 1, (3.44)

where h(t−ti) is the detector response due to a single incident photon leadingto a photo electron and therefore a photo current.If there is no correlation in the arrival time of the photons, the number

of photo electrons extracted from the photodetector in two different timeintervals that are at least one response time of the detector apart must beindependent from each other. This means that the current fluctuation spec-trum SInIn(f) for frequencies shorter than the inverse detector response time,τ pd, must be constant or "white",

SInIn(f) = const., for |f | << 1/τ pd (3.45)

because only then will the variance of the detected photons or generatedphoto electrons grow linearly with the total averaging time, i.e or inversebandwidth of the measurment. The detailed calculation is in the appendix,here we argue qualitatively.We obtain for the fluctuation in detected photon number sn(t) over a

measurement interval [t, t+ T ], by integrating over the noise current

sn(t) =1

e

t+TZt

In(t0)dt0 =

T

e

1

T

t+TZt

In(t0)dt0. (3.46)

As we know from 6.011, for a stationary random process In(t), the varianceif it is the same as the variance of sn and the variance can be computed fromthe autocorrelation spectrum according to

σ2snsn =s2n®=

T 2

e2

∆fZ−∆f

SInIn(f) · df (3.47)

=T 2

e2SInIn(f) · 2∆f. (3.48)


As in the case of averaging over many (infinitly many) modes in the previoussection, one can show that for a randomly arriving stream of particles, herephotons, which have no correlations with each other and a given average flux(Poisson process), the variance of the number of particles/photons measuredin a given time interval T should be equal to the average number of detectedparticles independent from the observation interval T

σ2snsn =T 2

e2SInIn(f)2∆f = hsi = I

eT, (3.49)

i.e. the power spectral density of the noise current is

SInIn(f) = eI1

2T∆f= eI, (3.50)

where we have used 2T∆f = 1 to derive at the correct noise formula forthe shot noise due to a poison distributed current, see appendix. Note, thatSInIn(f) is the two-sided autocorrelation spectrum, i.e. defined over positiveand negative frequencies. If only positive frequencies are used, we need tomulitply by a factor of two (SInIn(f) = 2eI).

One may think that shot noise is rather due to the discreteness of theelectrical charge than due to the presence of photons. However, this is notso. There exist optical sources that generate a photon stream which hasless noise than the shot noise formula 3.50 predicts, and which can not beexplained classically.

3.3.1 Signal to Noise Ratio

Thus if a constant optical power is applied to a photo detector the signal tonoise ratio in terms of signal power to noise power when measured with abandwdith ∆f is

SNR =I2

2SInIn(f)∆f=

I

2e∆f=

ηQEP

2hf∆f. (3.51)

The signal to noise ratio only growth linearly with the optical power, sincealso the noise grows with the applied optical power but less strong.

3.4. SPONTANEOUS AND INDUCED EMISSION 177

3.3.2 Noise Equivalent Power

Most often a real photo detector includes an additional amplification processwhich may add addtional noise. Then the signal to noise ratio changes toaccount for these additional noise sources. Manufacturers of photodetectorstherfore specify a noise equivalent power NEP which has the units [W/

√Hz].

It simply specifies the optical power level necessary on the photodetector toachieve a signal to noise ratio of 1.

3.4 Spontaneous and Induced Emission

The number of photons in a radiation mode may change via emission of aphoton into the mode or absorption of a photon from the mode by atoms,molecules or a solid state material. Einstein introduced a phenomenologicaltheory of these processes in order to explain how matter may get into thermalequilibrium by interaction with the modes of the radiation field. He consid-ered the interaction of a mode with atoms modeled by two energy levels E1and E2, see Figure 3.8.

Figure 3.8: Energy levels of a two level atom and populations.

n1 and n2 are the population densities of these two levels considering awhole ensemble of these atoms. Transitions are possible in the atom betweenthe two energy levels by emission of a photon at a frequency

f =E2 −E1

h(3.52)


Absorption of a photon is only possible if there is energy present in theradiation field. Einstein wrote for the corresponding transition rates, whichshould be proportional to the population densities and the photon density atthe transition frequency

−dn1dt

¯Abs

=dn2dt

¯Abs

= B12n1w(f21). (3.53)

The coefficient B12 characterizes the absorption properties of the transition.Einstein had to allow for two different kind of processes for reasons that be-come clear a little later. Transitions induced by the already present photonsor radiation energy as well as spontaneous transitions

dn1dt

¯Em

= −dn2dt

¯Em

=B21 n2w (f21) +A21 n2 . (3.54)

The coefficient B21describes the induced and A21 the spontaneous emissions.The latter transitions occur even in the absence of any radiation and thecorresponding coefficient determines the lifetime of the excited state

τ sp = A−121 , (3.55)

in the absence of the radiation field. The total change in the populationdensities is due to both absorption and emission processes

dnidt=

dnidt

¯Em

+dnidt

¯Abs

, for i = 1, 2 (3.56)

Using Eqs.(3.53) and (3.54) we find

−dn1dt

=dn2dt

= (B12 n1 −B21 n2) w (f21)−A21 n2. (3.57)

In thermal equilibrium the energy density of the radiation field must fulfillthe condition

w (f21) =A21/B12

n1/n2 −B21/B12, (3.58)

while the atomic ensemble itself should also be in thermal equilibrium whichagain should be described by the Boltzmann statistics, i.e. the ratio betweenthe population densities are determined by the Boltzmann factor

n2/n1 = exp

µ−E2 −E1

kT

¶. (3.59)

3.4. SPONTANEOUS AND INDUCED EMISSION 179

And with it the energy density of the radiation field must be

w (f21) =A21/B12

exp¡hf21kT

¢−B21/B12

. (3.60)

A comparison with Planck’s law, Eq.(3.24), gives

B21 = B12, (3.61)

and

A21 =8π hf321

c3B12. (3.62)

Clearly, without the spontaneous emission process it is impossible to arriveat Planck’s Law in equilibrium. The spectral energy density of the radiationfield can be rewritten with the average photon number in the modes at thetransition frequency f21 as

w (f21) =8π f221c3

hf21 hsi , with hsi = hsi = 1

exp hf21kT− 1

. (3.63)

Or we can write

w (f21) =A21B12

hsi .

With that relationship Eq.(3.54) can be rewritten as

dn1dt

¯Em

= −dn2dt

¯Em

=A21 n2 (hsi+ 1) , (3.64)

which indicates that the number of spontaneous emissions is equvalent to in-duced emissions caused by the presence of a single photon per mode. Havingidentified the coefficients describing the transition rates in the atom interact-ing with the field from equilibrium considerations, we can rewrite the rateequations also for the non equilibrium situation, because the coefficients areconstants depending only on the transition considered

dn1dt

= −dn2dt=1

τ sp[(n2−n1) hsi+ n2] . (3.65)

With each transition from the excited state of the atom to the ground statean emission of a photon goes along with it. From this, we obtain a changein the average photon number of the modes

d hsidt

= Vdn1dt

, (3.66)


which isd hsidt

=V

τ sp[(n2 − n1) hsi+ n2] . (3.67)

Again the first term describes the stimulated or induced processes and thesecond term the spontaneous processes. As we will see later, the stimulatedemission processes are coherent with the already present radiation field that isinducing the transitions. This is not so for the spontaneous emissions, whichadd noise to the already present field. For n1 > n2 the stimulated processeslead to a decrease in the photon number and the medium is absorbing. In thecase of inversion, n2 > n1, the photon number increases exponentially. Ac-cording to Eq.(3.59) inversion corresponds to a negative temperature, whichis an indication for a non equilibrium situation that can only be maintainedby additional means. It is impossible to achieve inversion by simple irradia-tion of the atoms with intense radiation. As we see from Eq.(3.67) in steadystate the ratio between excited state and ground state population is

n2n1=

hsihsi+ 1 , (3.68)

which at most approaches equal population for very large photon number.However, such a process can be exploited in a three or four level system, seeFigure 3.9, to achieve inversion.

Figure 3.9: Three level system: (a) in thermal equilibrium and (b) underoptical pumping at the transition frequency f31.

3.5. MATTER WAVES AND BOHR’S MODEL OF AN ATOM 181

By optical pumping population from the ground state can be transfered tothe excited level with energy E3. If there is a fast relaxation from this level tolevel E2, where level two in contrast has a long lifetime, it is conceivable thatan inversion between level E2 and E1 can build up. If inversion is achievedradiation at the frequency f21 is amplified.

3.5 Matter Waves and Bohr’s Model of anAtom

By systematic scattering experiments Ernest Rutherford showed in 1911, thatthe negative charges in an atom are homogenously distributed in contrast tothe positive charge which is concentrated in a small nucleus about 10,000times smaller than the atom itself. The nucleus also carries almost all of theatomic mass. Rutherford proposed a model of an atom where the electronscircle the nucleas similar to the planets circling the sun where the gravita-tional force is replaced by the Coulomb force between the electrons and thenucleus.

This model had many short comings. How was it possible that the elec-trons, which undergo acceleration on their trajectory around the nucleus, donot radiate according to classical electromagnetism, loose energy and finallyfall into the nucleus? Due to advances in optical instrumentation the lightemitted from thermally excited atomic vapors was known to be in the formof discrete lines. Balmer found in 1885 that these lines could be expressedby the rule

1

λ= RH

µ1

22− 1

n2

¶, with n = 3, 4, 5, ... (3.69)

where λ is the wavelength of light and RH = 10.968 · μm−1 is the Rydbergconstant for hydrogen. For n = 3 this corresponds to the red Hα-line atλ = 656.3nm, for n→∞ one obtains the wavelength of the limiting line inthis series at λ = 364.6nm, see Figure 3.10.


Figure 3.10: Balmer series on a wave number scale.

.

In the subsequent spectroscopy work further sequences where found:1. Lyman Series:

1

λ= RH

µ1

12− 1

n2

¶, with n = 2, 4, 5, ... (3.70)

2. Balmer Series:

1

λ= RH

µ1

22− 1

n2

¶, with n = 3, 4, 5, ... (3.71)

3. Paschen Series:

1

λ= RH

µ1

32− 1

n2

¶, with n = 4, 5, 6, ... (3.72)

4. Brackett Series:

1

λ= RH

µ1

42− 1

n2

¶, with n = 5, 6, 7, ... (3.73)

5. Pfund Series:

1

λ= RH

µ1

52− 1

n2

¶, with n = 6, 7, 8, ... (3.74)

The Lyman series in the UV-region of the spectrum, whereas the Pfund seriesis in the far infrared. These sequences can be represented as transitionsbetween energy levels as shown in Figure 3.11.


Figure 3.11: Energy level diagram for the hydrogen atom.

.

In 1913, Niels Bohr found the quantization condition for the electrontrajectories in the Hydrogen atom and he was able to derive from that thespectral series discussed above. He postulated that only those electron tra-jectories are allowed that within one rountrip around the nucleus have anaction equal to a multiple of Planck’s quantum of action h.I

p · ds = nh, with n = 1, 2, 3.... (3.75)

Second, he postulated that the electron can switch from one energy level ortrajectory to another one by the emission or absorption of a photon with an


energy equivalent to the energy difference between the two energy levels, seeFigure 3.12.

hf = ∆E. (3.76)

Figure 3.12: Transition between different energy levels in the hydrogen atom.

Assuming a circular trajectory of the electron with radius rn around thenucleus, the quantization condition for the electron trajectory (3.75) leads to

mvn · 2πrn = nh, with n = 1, 2, 3... (3.77)

The other condition for radius and velocity of the electron around the nucleusis given by the equality of Coulomb and centrifugal force at radius rn, whichleads to

e2

4πε0r2n=

mv2nrn

, (3.78)

or

v2n =e2

4πε0rnm. (3.79)

Substituting this value for the electron velocity in the squared quantizationcondition (3.77), we find the radius of the electron trajectories

rn =ε0h

2

πe2mn2. (3.80)

The radius of the first trajectory, called Bohr radius is r1 = 0.529 · 10−10m.The velocities on the individual trajectories are

vn =e2

2ε0h

1

n. (3.81)


The highest velocity is found for the tightest trajectory around the nucleus,i.e. for n = 1, which can be expressed in terms of the velocity of light as

v1 =e2

2ε0hc· c = 1

137· c, (3.82)

where e2

2ε0hc= 1

137is the fine structure constant.

The energy of the electrons on these trajectories with the quantum num-ber n is due to both potential and kinetic energy

Ekin =1

2mv2n =

me4

8ε20h2n2

, (3.83)

Epot = − e2

4πε0rn= − me4

4ε20h2n2

. (3.84)

or

En = Ekin +Epot (3.85)

En = − me4

8ε20h2n2

. (3.86)

Note, the energy of a bound electron is negative. For n→∞, En = 0. Theelectron becomes detached from the atom, i.e. the atom becomes ionized.The lowest and most stable energy state of the electron is for n = 1

E1 = −me4

8ε20h2= −13.53eV, (3.87)

with correspondes to the ground state in hydrogen. When a transition be-tween two of this energy eigenstates occurs a photon with the correspondingenergy is released

hf = Ek −En, (3.88)

= − me4

8ε20h2

µ1

k2− 1

n2

¶. (3.89)

3.6 Wave Particle Duality


Bohr’s postulates were not able to explain all the intricacies of the observedspectra and they couldn’t explain satisfactory the structure of the more com-plex atoms. This was only achieved with the introduction of wave mechanics.In 1923, de Broglie was the first to argue that matter might also have waveproperties. Starting from the equivalence principle of mass and energy byEinstein

E0 = m0 c20, (3.90)

he associated a frequency with this energy accordingly

f0 = m0 c20/h. (3.91)

Since energy and frequency are not relativistically invariant quantities butrather components of a four-vector which has the particle momentum as itsother components (E0/c0, px,py, pz) or (ω0/c0, kx,ky, kz), it was a necessitythat with the energy frequency relationship

E = hf = ~ω, (3.92)

there must also be a wave number associated with the momentum of a particleaccording to

p = ~k. (3.93)

In 1927, C. J. Davisson and L. H. Germer experimentally confirmed thisprediction by finding strong diffraction peaks when an electron beam pene-trated a thin metal film. The pictures were close to the observations of Lauein 1912 and Bragg in 1913, who studied the structure of crystaline and polycrystaline materials with x-ray diffraction.With that finding the duality between waves and particles for both light

and matter was established. Duality means that both light and matter havesimultaneous wave and particle properties and it depends on the experimentalarrangement whether one or the other property manifests itself strongly inthe experimental outcome.

3.7 Appendix A: Mode Counting

It is interested to estimate the number of modes one is averaging over givena certain emitting surface and a certain measurement time, see Figure 3.13.

3.7. APPENDIX A: MODE COUNTING 187

x

yz

Lx

Ly

Lz

As ADkAcΩc

Figure 3.13: Counting of longitudinal and transverse modes excited from aradiating surface of size As.

If the area As is emitting light, it will couple to the modes of the freefield. To count the modes we put a large box (universe) over the experi-mental arrangement under consideration. The emitting surface is one side ofthe box. The light from this surface, i.e. specifying the transverse electricand magnetic fields, couples to the modes of the universe with wave vectorsaccording to Eq.(3.5).

Longitudinal Modes

The number of longitudinal modes, that propagate along the positive z-direction in the frequency interval∆f can be derived from∆k = (2π/Lz)∆Nand ∆k = (2π/c0)∆f

∆N =Lz

c0∆f, (3.94)

or using the propagation or measurement time over which the experimentextends

τ = Lz/c0, (3.95)

we obtain for the number of longitudinal modes that are involved in themeasurement that is carried out over a time intervall τ and a frequencyrange ∆f

∆N = τ∆f . (3.96)


Transverse Modes

The free space modes that arrive at the detector area AD will not only havewave vectors with a z−component, but also transverse components. Lets as-sume that the detector area is far from the emitting surface, and we consideronly the paraxial plane waves. The wave vectors of these waves at a givenfrequency or free space wave number k0 can be approximated by

kmn =

µ2πm

Lx,2πn

Ly, k0

¶with m,n = 0, 1, 2, ... (3.97)

where m and n are transverse mode indices. Then one mode occupies thevolume angle

Ωc =4π2

LxLy k20,

= λ20/As . (3.98)

If the modes are thermally excited, the radiation in individual modes isuncorrelated. Therefore, if there is a detector at a distance r then onlythe field within an area

Ac = r2Ωc , (3.99)

is correlated. If the photodetector has an area Ad, then the number oftransverse modes detected is

Nt = Ad/Ac. (3.100)

The total number of modes detected is

Ntot =Ad

Acτ∆f =

AdAs

r2λ20τ∆f . (3.101)

Note, that there is perfect symmetry between the area of the emitting andreceiving surface. The emitter and the receiver could both be black bodies.If one of them is at a higher temperature than the other, there is a net flow ofenergy from the warmer body to the colder body until equilibrium is reached.This would not be possible without interaction over the same number ofmodes. Thus the formula which is completely unrelated to thermodynamicsis necessary to fulfill one of the main theorems of thermodynamcis, that isthat energy flows from warmer to colder bodies.

3.8. APPENDIX B: SHOT NOISE FORMULA 189

3.8 Appendix B: Shot Noise Formula

We obtain for the deviation in detected photon number sn over a measure-ment time T

sn(t) =

Z t+T

t

In(t0)

edt0 (3.102)

which can be expressed as a linear filtering problem with the rectangularfunction G(t)

sn(t) =T

eIn(t) ∗G(t) (3.103)

where

G(f) =1

j2πfT(ejπfT − e−jπfT ) =

sin(πfT )

(πfT ), (3.104)

As we know from 6.011 the autocorrelation spectrum of a stationary randomprocess at the output of a linear filter is given by

Ssnsn(f) =T 2

e2SInIn(f) ·

¯G(f)

¯2(3.105)

and its variance is given by the integral over the autocorrelation spectrum

σ2snsn =s2n®=

T 2

e2

∞Z−∞

SInIn(f) ·¯G(f)

¯2df (3.106)

=T 2

e2SInIn(f) ·

∞Z−∞

¯G(f)

¯2df =

T 2

e2SInIn(f)

1

T. (3.107)

This variance should be equal to the total number of detected photons inde-pendent from the observation interval T

σ2snsn =T 2

e2SInIn(f)

1

T= hsi = I

eT, (3.108)

orSInIn(f) = eI. (3.109)

Bibliography

[1] Fundamentals of Photonics, B.E.A. Saleh and M.C. Teich,

[2] Optical Electronics, A. Yariv, Holt, Rinehart & Winston, New York,1991.

[3] Introduction to Quantum Mechanics, Griffiths, David J., Prentice Hall,1995.

[4] QuantumMechanics I, C. Cohen-Tannoudji, B. Diu, F. Laloe, JohnWileyand Sons, Inc., 1978.

191

192 BIBLIOGRAPHY

Chapter 4

Schroedinger Equation

Einstein’s relation between particle energy and frequency Eq.(3.92) and deBroglie’s relation between particle momentum and wave number of a corre-sponding matter wave Eq.(3.93) suggest a wave equation for matter waves.This search for an equation describing matter waves was carried out by ErwinSchroedinger. He was successful in the year 1926.The energy of a classical, nonrelativistic particle with momentum p that

is subject to a conservative force derived from a potential V (r) is

E =p 2

2m+ V (r) . (4.1)

For simplicity lets begin first with a constant potential V (r) = V0 = const.This is the force free case. According to Einstein and de Broglie, the dis-persion relation between ω and k for waves describing the particle motionshould be

~ω =~2k2

2m+ V0. (4.2)

Note, so far we had a dispersion relation for waves in one dimension, wherethe wavenumber k(ω), was a function of frequency. For waves in three dimen-sions the frequency of the wave is rather a function of the three componentsof the wave vector. Each wave with a given wave vector k has the followingtime dependence

ej(k·r−ωt), with ω =~k2

2m+

V0~

(4.3)

Note, this is a wave with phase fronts traveling to the right. In contrast to ournotation used in chapter 2 for rightward traveling electromagnetic waves, we

193

194 CHAPTER 4. SCHROEDINGER EQUATION

switched the sign in the exponent. This notation conforms with the physicsoriented literature. A superposition of such waves in k−space enables us toconstruct wave packets in real space

Ψ (r, t) =

Zφω

³k, ω

´ej(k·r−ωt)d3k dω (4.4)

The inverse transform of the above expression is

φω

³k, ω

´=

1

(2π)4

ZΨ (r, t) e−j(k·r−ωt)d3r dt, (4.5)

with

φω

³k, ω

´= φ (k) δ

Ãω − ~k

2

2m− V0~

!. (4.6)

Or we can rewrite the wave function in Eq.(4.4) by carrying out the trivialfrequency integration over ω

Ψ (r, t) =

Zφ (k) exp

Ãj

"k·r −

Ã~k2

2m+

V0~

!t

#!d3k. (4.7)

Due to the Fourier relationship between the wave function in space and timecoordinates and the wave function in wave vector and frequency coordinates

φω

³k, ω

´↔ Ψ (r, t) (4.8)

we have

ω φω (k, ω) ↔ j∂Ψ (r, t)

∂t, (4.9)

k φω (k, ω) ↔ − j∇Ψ (r, t) , (4.10)

k2 φω (k, ω) ↔ −∆ Ψ (r, t) . (4.11)

where

∇ = ex∂

∂x+ ey

∂

∂y+ ez

∂

∂z, (4.12)

∆ = ∇ ·∇ ≡ ∇2 = ∂2

∂x2+

∂2

∂y2+

∂2

∂z2. (4.13)

195

From the dispersion relation follows by multiplication with the wave functionin the wave vector and frequency domain

~ω φω (k, ω) =~2k2

2mφω (k, ω) + V0 φω (k, ω) . (4.14)

With the inverse transformation the corresponding equation in the space andtieme domain is

j ~∂Ψ (r, t)

∂t= − ~

2

2m∆ Ψ (r, t) + V0 Ψ (r, t) . (4.15)

Generalization of the above equation for a constant potential to the instanceof an arbitrary potential in space leads finally to the Schroedinger equation

j ~∂Ψ (r, t)

∂t= − ~

2

2m∆ Ψ (r, t) + V (r) Ψ (r, t) . (4.16)

Note, the last few pages ar not a derivation of the Schroedinger Equationbut rather a motivation for it based on the findings of Einstein and deBroglie.The Schroedinger Equation can not be derived from classical mechanics. Butclassical mechanics can be rederived from the Schroedinger Equation in somelimit. It is the success of this equation in describing the experimentally ob-served quantummechanical phenomena correctly, that justifies this equation.The wave function Ψ (r, t) is complex. Note, we will no longer underline

complex quantities. Which quantities are complex will be determined fromthe context.Initially the magnitude square of the wave function |Ψ (r, t)|2 was inter-

preted as the particle density. However, Eq.(4.15) in one spatial dimensionis mathematical equivalent to the dispersive wave motion Eq.(2.77), wherethe space and time variables have been exchanged. The dispersion leads tospreading of the wave function. This would mean that any initially compactparticle, which has a well localized particle density, would decay, which doesnot agree with observations. In the framwork of the "Kopenhagen Interpre-tation" of Quantum Mechanics, whose meaning we will define later in detail,|Ψ (r, t)|2 dV is the probability to find a particle in the volume dV at positionr , if an optimum measurement of the particle position is carried out at timet. The particle is assumed to be point like. Ψ (r, t) itself is then consideredto be the probability amplitude to find the particle at position r at time t.Note, that the measurement of physical observables like the position of

a particle plays a central role in quantum theory. In contrast to classical


mechanics where the state of a particle is precisely described by its positionand momentum in quantum theory the full information about a particle isrepresented by its wave function Ψ (r, t). Ψ (r, t) enables to compute theoutcome of a measurement of any possible observable related to the particle,like its position or momentum.Before, we discuss this issue in more detail lets look at a few examples to

get familiar with the mathematics of quantum mechanics.

4.1 Free Motion

Eq.(4.15) describes the motion of a free particle. For simplicity, we consideronly a one-dimensional motion along the x-axis. Initially, we might onlyknow the position of the particle with finite precision and therefore we use aGaussian wave packet with finite width as the initial wave function

Ψ (x, t = 0) = A exp

µ− x2

4σ20+ jk0x

¶. (4.17)

The probability density to find the particle at position x is a Gaussian dis-tribution

|Ψ (x, t = 0)|2 = |A |2 expµ− x2

2σ20

¶, (4.18)

σ20 is the variance of the initial particle position. Since the probability to findthe particle somewhere must be one, we can determine the amplitude of thewave function by requiring

∞Z−∞

|Ψ (x, t = 0)|2 dx = 1→ A =1

4√2π√σ0

(4.19)

The meaning of the wave number k0 in the wave function (4.17) becomes ob-vious by expressing the solution to the Schroedinger Equation by its Fouriertransform

Ψ (x, t) =

+∞Z−∞

φ (k) exp [ j (kx− ω (k) t)] dk (4.20)

4.1. FREE MOTION 197

or specifically for t = 0

Ψ (x,0) =

+∞Z−∞

φ (k) e jkx dk , (4.21)

or

φ (k) =1

2π

+∞Z−∞

Ψ(x,0) e− jkx dx . (4.22)

For the initial Gaussian wavepacket of

Ψ (x, 0) = A exp

µ− x2

4σ20+ jk0x

¶(4.23)

we obtainφ (k) =

Aσ0√πexp

£−σ20 (k − k0)

2¤ . (4.24)

This is a Gaussian distribution for the wave number, and therefore momen-tum, of the particle with its center at k0. With the dispersion relation

ω =~ k2

2 m, (4.25)

with the constant potential V0 set to zero, the wave function at any latertime is

Ψ (x, t) =Aσ0√π

+∞Z−∞

exp

∙−σ20 (k − k0)

2− j~k2

2mt+ jkx

¸dk. (4.26)

This is exactly the same Gaussian integral we were studying for dispersivepulse propagation or the diffraction of a Gaussian beam in chapter 2 whichresults in

Ψ (x, t) =Aq

1 + j ~ tm2σ20

exp

⎡⎣− x2 − 4 jσ20k0 x+ j~2σ20k20

mt

4σ20

³1 + j ~

m2σ20t´

⎤⎦ . (4.27)

As expected the wave packet stays Gaussian. The probability density is

|Ψ(x, t)|2 = |A|2q1 + ( ~ t

2mσ20))exp

⎡⎢⎢⎣−¡x− ~ k0

mt¢2

2σ20

∙1 +

³~ t2mσ20

´2¸⎤⎥⎥⎦ . (4.28)


With the value for the amplitude A according to Eq.(4.19), the wave packetremains normalized

+∞Z−∞

|Ψ(x, t)|2 dx = 1. (4.29)

Using the probability distribution for the particle position, we obtain for itsexpected value

hxi =+∞Z−∞

x |Ψ(x, t)|2 dx (4.30)

or

hxi = ~ k0m

t . (4.31)

Thus the center of the wave packet moves with the velocity of the classicalparticle

υ0 =~ k0m

, (4.32)

which is the group velocity derived from the dispersion relation (4.2)

υ0 =∂ω(k)

∂k

¯k=k0

=1

~∂E(k)

∂k

¯k=k0

. (4.33)

As we will see later, the expected value for the center of mass of the par-ticle follows Newton’s law, which is called Ehrenfest’s Theorem. For theuncertainty in the particle position

∆x =

qhx2i− hxi2 (4.34)

follows for the freely moving particle

∆x = σ0

s1 +

µ~ t2mσ20

¶2. (4.35)

The probability density for the particle position disperses over time. Asymp-totically one finds

∆x.=

~ t2mσ20

for~ t2mσ20

À 1 . (4.36)

Figure 4.1 (a) is a sketch of the complex wave packet and (b) indicates thetemporal evolution of the average and variance of the particle center of mass

4.1. FREE MOTION 199

motion described by the complex wave packet. The wave packet dispersesfaster, if it is initially stronger localised.

Figure 4.1: Gaussian wave packet: (a) Real and Imaginary part of the com-plex wave packet. (b) width and center of mass of the wave packet.

Example:Using this one dimensional model, we can estimate how rapidly an elec-

tron moves in a hydrogen atom. If we localize an electron in a box with a sizesimilar to that of a hydrogen atom, i.e. σ0 = a0 = 0.5 · 10−10m, without thepresence of the proton that holds the electron back from escaping, it will onlytake t = 2mσ20/~ = 2 ·9.81 ·10−31kg· (0.5 · 10−10)

2m2/6.626 ·10−34Js = 46.5as(attoseconds=10−18 sec) until its wave function disperses significantly. Thisresult indicates that electronic motion in atoms occurs on a attosecond timescale. Note, these time scales quickly become very long if macroscopic ob-jects are described quantum mechanically. For example, for a particle with


a mass of 1μg localized in a box with dimensions 1μm, the equivalent timefor significant dispersion of the wave function is t = 2 · 1019s = 2Millionyears. This result gives us a first indication why we are far, far away fromencountering quantum mechanical effects in our everyday life and why themechanics of the micro cosmos, on an atomic or molecular level, is so dif-ferent from our macroscopic experience. The reason is the smallness of thequantum of action h.The reason for this behaviour is that a well localized particle has a wider

momentum or wave number distribution. This is in one to one analogy thatan otpical pulse disperses faster in a medium with a given dispersion if it isshorter because of larger spectral width. The wave number spread is

(∆k)2 =

∞Z−∞

(k − k0)2 |φ (k)|2 dk

∞Z−∞

|φ (k)|2 dk

. (4.37)

Here, we have

∆k =1

2σ0, (4.38)

or for the momentum spread

∆p =~2σ0

. (4.39)

The position-momentum uncertainty product is then

∆p ∆x =~2

s1 +

µ~ t2m2

0

¶2. (4.40)

The position-momentum uncertainty product is a minimum at t = 0 andsteadily increases from this initial value. As we will show later it is in gen-erally true that the position-momentum uncertainty product satisfies thecondition

∆xi∆pi >~2. (4.41)

Note, that the index i indicates the coordinate x, y or z. This is Heisenberg’suncertainy relation between particle position and moment, which holds for

4.2. PROBABILITYCONSERVATIONANDPROPABILITYCURRENTS201

each component individually. Later, we will find other pairs of physicalobservavles, which are called conjugate observables and which satisfy similaruncertainty relations. The product of such quantities is always an action.This is for example also true for the product of energy and frequency andthe resulting energy-time uncertainty relation is

∆E∆t > ~2. (4.42)

Note, whereas the position-momentum uncertainy is related to the choice ofthe particle state described by the wave function, the energy-time uncertaintyrelation is related to the dynamics of a quantum process. What it means isthat a quantum system can only change its state significantly within a timespan ∆t, if the state, the quantum system is in, has an energy uncertaintylarger than δE > ~

2δt.

Position and momentum variables that do not belong to the same degreeof freedom, such as y, and px are not subject to an uncertainty relation.

4.2 Probability Conservation and Propabil-ity Currents

Max Born was the first to introduce the propabilistic interpretation of thewave function found by Schroedinger, that is the propability to find the centerof mass of a particle at position r in a volume element dV is given by themagnitude square of the wave function multiplied by dV

p (r, t) = |Ψ (r, t)|2 dV . (4.43)

If this interpretation makes sense, then the total propability that the parti-cle can be found somewhere should by 1 and this normalization should notchange during the dynamics. We found that this is true for the Gaussianwave packet undergoing free motion. Here, we want to show that this is trueunder the most general circumstances. We look at the rate of change of the


probability to find the particle in an arbitrary but fixed volume V = V ol

d

dt

ZV ol

p (r, t) d3r = (4.44)

=

ZV ol

∂

dt|Ψ (r, t)|2 d3r

=

ZV ol

∙µ∂

∂tΨ∗ (r, t)

¶Ψ (r, t) +Ψ∗ (r, t)

µ∂

∂tΨ (r, t)

¶ ¸d3r

Using the Schroedinger Equation (4.16) for the temporal change of the wavefunction we obtain

d

dt

ZV ol

p (r, t) d3r =

=

ZV ol

∙µ~

j2m∇ ·∇ Ψ∗ (r, t)− j

~V (r) ∗Ψ∗ (r, t)

¶Ψ (r, t)

¸d3r (4.45)

+

ZV ol

∙Ψ∗ (r, t)

µ− ~j2m∇ ·∇ Ψ (r, t) +

j

~V (r) Ψ (r, t)

¶¸d3r

Since the potential V (r) is real the terms related to it cancel. The other twoterms can be written as the divergence of a current densityZ

V ol

∂

∂tp (r, t) d3r = −

ZV ol

∇ · J (r, t) d3r, (4.46)

with

J (r, t) =~

j2m(Ψ∗ (r, t) (∇Ψ (r, t))−Ψ (r, t) (∇Ψ∗ (r, t))) . (4.47)

Eq.(4.46) is true for any volume, i.e.ZV ol

∙∂

∂tp (r, t) +∇ · J (r, t)

¸d3r = 0, (4.48)

which is only possible if the integrand vanishes

∂

∂tp (r, t) = − ∇ · J (r, t) . (4.49)

Clearly, J (r, t) has the physical meaning of a probability current. The prob-ability in a volume element changes because of probablity flowing out of

4.2. PROBABILITYCONSERVATIONANDPROPABILITYCURRENTS203

this volume element. Note, this is the same local law that we have for theconservation of charge. In fact, if the particle is a charged particle, like anelectron is, multiplication of J with −e0 would result in the electrical currentassociated with the wave function Ψ (r, t) .Gauss’s theorem statesZ

V ol

∇ · J (r, t) d3r =

ZS

J (r, t) dS, (4.50)

where V ol is the volume over which the integration is carried out and S isthe surface that encloses the volume with dS an outward pointing surfacenormal vector. With Gauss’s theorem the local conservation of probabilitycan be transfered to a global result, since

d

dt

ZV ol

p (r, t) d3r = −ZV ol

∇ · J (r, t) d3r = −ZS

J (r, t) dS. (4.51)

If we choose as the volume the whole space and if Ψ (r, t) and ∂∂tΨ (r, t)

vanish rapidly enough for r →∞ such that the probability current vanishesat infinity, the total probability is conserved. These findings proove thatthe probabilty interpretation of the wave function is a valid interpretationnot contradicting basic laws of probability. If the wave function properlynormalized at the beginning it will stay normalized.

Example The Gaussian wave packet satisfies the condition that the prob-ability current decays rapidly enough at the surface of a large enough chosenvolume so that the normalization is preserved. A monochromatic plane wavedoes not satisfy this condition. However, the probability current density givesa physical meaning to it. The wave function corresponding to a plan wave

Ψ (r, t) = ej(k·r−ωt), with ω =~k2

2m+

V0~

(4.52)

which is not normalizable, results in a homogenous probability current

J (r, t) =~

j2m(Ψ∗ (r, t) (∇Ψ (r, t))−Ψ (r, t) (∇Ψ∗ (r, t))) (4.53)

=~km|Ψ (r, t)|2 = p

m|Ψ (r, t)|2 = v · |Ψ (r, t)|2 ,


that is identical to the classical velocity of the particle. Thus a plane wavedescribes a particle with a precise velocity or momentum but completelyunknown position, therefore the related probability current density is com-pletely homogenous but directed into the direction of v. Such waves describethe initial state in a scattering experiment, where we shoot particles with aprecisely defined velocity v or momentum p or energy E = mv2

2= ~p2

2m= ~k2

2m

onto another object described by a scattering potential, see problem set. Theposition of these particles is completely unspecified, i.e. |Ψ (r, t)|2 =const.

4.3 Measureability of Physical Quantities (Ob-servables)

The reason for the more intricate description necessary for microscopic pro-cesses in comparison with macroscopic processes is simply the fact that thesesystems are so small that the interaction of the system with an eventualmeasurement apparatus can no longer be neglected. It turns out this factis not to overcome by choosing more and more sophisticated measurementapparati but rather is a principle limitation. If this is so, then it eventuallydoesn’t make sense or it becomes even inconsistent to attribute to a systemmore precisely defined physical quantities than actually can be retrieved bymeasurements. This is the physical reason behind the introduction of thewave function instead of the precisely defined position and momentum of theparticle that we used to deterministically predict the trajectory of a particlein an external field.It is impossible to assign to a microscopic particle a precise position and

momentum at the same time. To demonstrate this, we consider the following(Heisenberg) microscope to measure the exact position of a particle. We uselight with wavelength λ and focus it strongly with a lense of some focaldistance d, see Figure 4.2.From our construction of the Gaussian beam in section 2.5.2, we found

that if we generate a focused beam with a waist wo having a Rayleigh rangezR =

πw2oλ, the beam is composed of plane waves which have a Gaussian distri-

bution in its transverse k-vector, which has a variance k2T/2, see Eq.(2.268).

σk =k2T2

(4.54)

The Rayleigh range of the beam is related to the transverse wave number

4.3. MEASUREABILITYOFPHYSICALQUANTITIES (OBSERVABLES)205

spread of the beam by zR = k0/k2T , with k0 = 2π/λ, see (2.269) and there-

after. Note, the intensity profile of the beam has a variance w2o /4. If a particlecrosses the focus of the beam and scatters a single photon, which we detectwith the surrounding photo detector arrangement, then it is reasonable toassume that we know the position of the particle in the x-direction, with anuncertainty equal to the uncertainty in the transverse photon or intensitydistribution of the beam, i.e. ∆x = wo /2.

Photodector

Weak particle beamwith precise momentum p

p

Figure 4.2: Determination of particle position with an optical microscope.A weak particle beam with precisely defined moment p of the particles isdirected towards the focus of the Gaussian beam. In the focus the particlescatters at least one photon. Detection of the scattered photon with thesurrounding photodetector signals, that the position of the particle in x-direction has been determined within the beam waist of the Gaussian beam.However, due to the scattering of the photon a momentum uncertainty hasbeen introduced to the particle state.

During the measurement, the photon recoil induces a momentum kickwith an uncertainty∆px = ~kT/

√2. So even if the momentum of the particle

was perfectly know before the measurement, after the additional determina-tion of its position with a precision ∆x it has at least aquired an uncertaintyin its momentum of magnitude ∆px. The product of the uncertainties inpostion and momentum after the measurement is


∆px ·∆x =~kT√2· wo

2=~wo· wo

2=~2. (4.55)

Note, this result is exact and is independent of focusing. Tighter focusingwill enable us to more precisely determine the position of the particle, butwe will introduce more momentum uncertainty due to the photon recoil; theopposite is true for less focusing. Since we can not determine, and therefore,prepare a particle in a state with its position and momentum more preciselydetermined than this uncertainty product allows, there is no such state and(4.55) is the minimum uncertainty product achievable.The experimental setup can easily be extended to measure the momentum

and position of a particle in all three dimensions. For example one can usethree focused laser beams at different wavelength, which are orthogonal toeach other. Once a particle will fly through the focus and scatters threephotons, each of different color. If we knew its momentum initially precisely,we would know afterwards its 3-dimensional position with a position andmomentum spread as described by Eq.(4.55).

4.4 Stationary States

One of the great mysteries before the advent of quantum mechanics was theorgin of the discrete energy spectra observed in spectroscopic investigationsand empirically described by the Bohr-Sommerfeld model of the atom. Thismystery is easily explained by the Schroedinger Equation (4.16)

j~∂Ψ (r, t)

∂t= − ~

2

2m∆ Ψ (r, t) + V (r) Ψ (r, t) . (4.56)

It allows for solutionsΨ (r, t) = ψ (r) e−jωt, (4.57)

which have a time independent probability density, i.e.

|Ψ (r, t)|2 = |ψ (r)|2 = const., (4.58)

which is the reason for calling these states stationary states. Since the rightside of the Schroedinger Equation is equal to the total energy of the sys-tem, these states correspond to energy eigenstates of the system with energyeigenvalues

E = ~ω. (4.59)

4.4. STATIONARY STATES 207

These energy eigenstates ψ (r) are eigen solutions to the stationary or timeindependent Schroedinger Equation

− ~2

2m∆ ψ (r) + V (r) ψ (r) = E ψ (r) . (4.60)

We get familiar with this equation by considering a few one-dimensionalexamples, before we apply it to the Hydrogen atom.

4.4.1 The One-dimensional Infinite Box Potential

A simple example for a quantum mechanical system is an electron that canfreely move in one dimension x but only over a finite distance a. Such asituation closely describes an electron that is strongly bound to a moleculewith a cigar like shape with length a. The potential describing this situationis the one-dimensional box potential

V (x) =

½0, for |x| < a/2∞, for |x| ≥ a/2

, (4.61)

see Figure 4.3.

Figure 4.3: One dimensional box potential with infinite barriers.

In the interval [−a/2, a/2] the stationary Schroedinger equation is

−~2 d2ψ (x)

2m dx2= E ψ (x) . (4.62)


For |x| ≥ a/2 the wave function must vanish, otherwise the energy eigenvaluecan not be finite, i.e. ψ (x = ±a/2) = 0. This is analogous to the electricfield solutions for the TE-modes for a planar mirror waveguide and we find

ψn (x) =

r2

acos

nπx

afor n = 1, 3, 5 . . . , (4.63)

ψn (x) =

r2

asin

nπx

afor n = 2, 4, 6 . . . . (4.64)

The corresponding energy eigenvalues are

En =n2π2~2

2ma2. (4.65)

We also find that the stationary states constitute an orthogonal system offunctions

+∞Z−∞

ψm (x)∗ ψn (x) dx = δmn. (4.66)

In fact this system is complete. Any function in the interval [−a/2, a/2]can be expanded in a superposition of the basis functions ψn (x), which is aFourier series

f (x) =∞Xn=0

cnψn (x) (4.67)

with

cm =

Z a/2

−a/2ψm (x)

∗ f (x) dx, (4.68)

which is a consequence of the orthogonality relation (4.66).

Example: If we approximate the binding potential of a hydrogen atomby a one-dimensional box potential with a width equal to twice the Bohrradius a = 2a0 = 10−10m, the energy eigenvalues are En = n2 · 35eV. Clearly,the spacing of the energy eigenvalues does not conform with what has beenobserved experimentaly, compare with section 3.5, however the energy scaleis within an order of magnitude. The ionization potential of the hydrogenatom is 13.5eV .


4.4.2 The One-dimensional Harmonic Oscillator

The most important example of a quantum system is the one-dimensionalharmonic oscillator. It is the most basic mechanical and electrical systemand it describes the dynamics of a mode of the radiation field, see Figure 4.4.

Figure 4.4: Elastically bound particle

Mechanically, a harmonic oscillation comes about by the elastic forceobeying Hook’s law

F (x) = −Kx, (4.69)

that pulls back a particle with mass m in its equilibrium position. This forceis conservative and can be derived from a potential by

F (x) = −d V (x)

dx, (4.70)

withV (x) =

1

2Kx2 . (4.71)

Newton’s law results in the classical equation of motion

mx = F (x) , (4.72)

orx+ ω20x = 0, (4.73)

with the oscillation frequency

ω0 =

rK

m(4.74)


The corresponding stationary Schroedinger Equation is

d2ψ (x)

dx2+2m

~2

µE − 1

2Kx2

¶ψ (x) = 0. (4.75)

This equation is well known in mathematical physics and we want to bringit into standardized form by the scale transformation, i.e. introducing anormalized distance

ξ = ax, (4.76)

with the scale factor

a =

µmK

~2

¶ 14

=

rω0m

~=

rK

~ω0. (4.77)

In addition we introduce the energy scale factor

γ =2E

~ω0. (4.78)

Then the stationary Schroedinger Equation for the harmonic oscillator is

d2ψ (ξ)

dξ2+¡γ − ξ2

¢ψ (ξ) = 0. (4.79)

It turns out [1][6], that this equation has only solutions that are bounded,i.e. ψ (ξ → ±∞) = 0, if the normalized energies are

γn = 2n+ 1. (4.80)

And the corresponding eigensolutions are the Hermite Gaussians,

ψn (ξ) = const. Hn (ξ) e−12ξ2, (4.81)

which we discovered already as solutions of the paraxial wave equation, seeEqs.(2.312) and (2.313), i.e.

Hn (ξ) = (−1)n eξ2 dn

dξne−ξ

2

(4.82)

H0 (ξ) = 1 , H3 (ξ) = 8 ξ3 − 12 ξ ,

H1 (ξ) = 2 ξ , H4 (ξ) = 16 ξ4 − 48 ξ2 + 12 ,

H2 (ξ) = 4 ξ2 − 2 , H5 (ξ) = 32 ξ

5 − 160 ξ3 + 120 ξ .(4.83)


After denormalization of x and normalization of the stationary wave functionsare

ψn (x) =

ra

2n√π n!

Hn (ax) e−12a2x2. (4.84)

Again, we find that the Hermite Gaussians constitute an orthogonal systemof functions such that

+∞Z−∞

ψm (x)∗ ψn (x) dx = δmn. (4.85)

Figure 4.5 shows the first six stationary states or energy eigenstates of theharmonic oscillator.

Figure 4.5: First six stationary states of the harmonic oscillators.

The energy eigenvalues of the stationary states are

En =

µn+

1

2

¶~ω0 . (4.86)

Note, that the energy eigenvalues are equidistant and the difference betweentwo energy eigenstates follows the findings of Planck. An oscillator has dis-crete energy levels which differ by energy quanta of size ~ω0, see Figure 4.6.


Figure 4.6: Lowest order wavefunctions of the harmonic oscillator and thecorresponding energy eigenvalues [3].

The only difference is, that the whole energy scale is shifted by the energyof half a quantum, which is the lowest energy eigenvalue. Thus the minimumenergy, or ground state energy, of a harmonic oscillator is not zero but E0 =12~ω0.It is obvious, that an oscillator can not have zero energy because itsenergy is made up of kinetic and potential energy

E =p2

2m+1

2Kx2. (4.87)

Since every state has to fulfill Heisenberg’s uncertainty relation ∆p ·∆x ≥ ~2,

one can show that the state with minimum energy possible has an energyE0 =

12~ω0, which is true for the ground state ψ0 (x) according to Eq.(4.84).

The stationary states of the harmonic oscillator correspond to states withprecisely definied energy but completely undefined phase. If we assume aclassical harmonic oscillator with a well defined energy E = 1

2Kx20. Note,

that during a harmonic oscillation the energy is periodically converted frompotential energy to kinetic energy. Then the oscillator oscillates with a fixedampltiude x0

x(t) = x0 cos (ω0t+ ϕ) . (4.88)

If the phase is assumed to be random in the interval [-π, π], one finds for the

4.5. THE HYDROGEN ATOM 213

probability density of the position x to be

p (x) = p(ϕ)

¯dϕ

dx

¯· 2 = 2

2π

1

x0 sin (ω0t+ ϕ)

=1

πpx20 − x2

. (4.89)

Figure 4.7 shows this probability density corresponding to an energy eigen-state ψn (x) with quantum large quantum number n = 10.

Figure 4.7: Probability density |ψ10|2 of the harmonic oscillator containingexactly 10 energy quanta.

On average, the quantum mechanical probability density agrees withthe classical probability density, which is some form of the correspondenceprinciple, which says that for large quantum numbers n the wave functionsresume classical properties.

4.5 The Hydrogen Atom

The simplest of all atoms is the Hydrogen atom, which is made up of apositively charged proton with rest mass mp = 1.6726231 × 10−27 kg, anda negatively charged electron with rest mass me = 9.1093897 × 10−31 kg.Therefore, the hydrogen atom is the only atom which consists of only two


Figure 4.8: Bohr Sommerfeld model of the Hydrogen atom.

particles. This makes an analytical solution of both the classical as well asthe quantum mechanical dynamics of the hydrogen atom possible. All otheratomes are composed of a nucleus and more than one electron. Accordingto the Bohr-Somerfeld model of hydrogen, the electron circles the proton ona planetary like orbit, see Figure 4.8.The stationary Schroedinger Equationfor the Hydrogen atom is

∆ψ (r) +2m0

~2(E − V (r)) ψ (r) = 0 (4.90)

The potential is a Coulomb potential between the proton and the electronsuch that

V (r) = − e204π ε0 |r|

(4.91)

and the mass is actually the reduced mass

m0 =mp · me

mp +me(4.92)

that arises when we transform the two body problem between electron andproton into a problem for the center of mass and relative coordinate motion.Due to the large, but finite, mass of the proton, i.e. the proton mass is 1836times the electron mass, both bodies circle around a common center of mass.The center of mass is very close to the position of the proton and the reducedmass is almost identical to the proton mass.Due to the spherical symmetry of the potential the use of spherical co-

ordinates is advantageous. The Laplace operator in spherical coordinates


is

∆ψ =∂2ψ

∂r2+2

r

∂ψ

∂r+1

r2

∙1

sinϑ

∂

∂ϑ

µsinϑ

∂ψ

∂ϑ

¶+

1

sin2 ϑ

∂2ψ

∂ ϕ2

¸, (4.93)

and the stationary Schroedinger Equation takes on the form

∂2ψ

∂r2+2

r

∂ψ

∂r+1

r2

∙1

sinϑ

∂

∂ϑ

µsinϑ

∂ψ

∂ϑ

¶+

1

sin2 ϑ

∂2ψ

∂ ϕ2

¸+2m0

~2

µE +

e204π ε0 r

¶ψ = 0.

(4.94)

4.5.1 Ground State

Before, we look at the general solutions for the stationary Schroedinger Equa-tion for the hydrogen atom, lets find the most simple solution. We look firstfor a radially symmetric solution ψ (r, ϑ, ϕ) = ψ1 (r) , which leads to

∙∂2

∂r2+2

r

∂

∂r

¸ψ1 (r) +

2m0

~2

µE +

e204π ε0 r

¶ψ1 (r) = 0. (4.95)

Asymptotically for large r, this equation approaches the form

∂2

∂r2ψ1 (r) +

2m0

~2E ψ1 (r) = 0. (4.96)

Since, the energy eigenvalues E, which we expect, will be negative, the solu-tion of this equation is an exponential and we choose a decaying exponentialthat will lead to a normalizable wave function for a bound electron.

ψ1 (r) = e−αr, with α =

r−2m0E

~2(4.97)

Substituting this trial solution into 4.96, we find in addtion

1

r

µ−2α+ m0e

20

2π ε0 ~2

¶ψ1 (r) = 0. (4.98)

This equation is only fulfilled for all r if

α =πm0e

20

ε0 h2= r−11 , (4.99)


where r1 = ε0h2

πe2m0≈ 0.529 · 10−10m, is the Bohr radius from Bohr’s models

of the atom, see Eq.(3.80). And the energy of the ground state follows from(4.97) to be

E1=~2

2m0r21= − me4

8ε20h2= −13.53eV (4.100)

in agreement with (3.87). We finally normalize the wave function, i.e. weintroduced a normalization constant C

ψ1 (r) = C · e−αr, (4.101)

determined such that

Z ∞

0

|ψ1 (r)|2 4πr2dr = 1, (4.102)

which leads to

C =

s1

πr31. (4.103)

Thus in total the properly normalized groundstate wave function of hyrdogenis

ψ1 (r) =

s1

πr31e−r/r1, (4.104)

Figure 4.9 shows the magnitude square of the hydrogen ground statewave function |ψ1 (r)|2 and the corresponding propability density to find theelectron in a radial interval [r, r + dr].


1

rr1

Figure 4.9: Sketch of the magnitude square of the hydrogen ground state.wave function and of the propability density to find an electron in a sphericalshell with radius r and thickness dr.


4.5.2 Excited States

To find the excites states of the hydrogen atom, we need to allow for a moregeneral solution than a spherical symmetric solution. We find a general solu-tion of the stationary Schroedinger Equation by using separation of variables

ψ (r, ϑ, ϕ) = R (r) · Y (ϑ, ϕ) . (4.105)

Substituting this trial wave function into the Schroedinger Equation using

d2R

dr2+2

r

dR

dr=1

r2∂

∂r

µr2

∂

∂rR

¶,

multiplying by r2 sin2 ϑ and deviding by R (r) Y (ϑ, ϕ) , we obtain from 4.93½1

R

∂

∂r

µr2

∂

∂rR

¶+2m0r

2

~2

µE +

e204π ε0r

¶¾=

− 1Y

∙1

sinϑ

∂

∂ϑ

µsinϑ

∂Y

∂ϑ

¶¸+

1

sin2 ϑ

∂2Y

∂ ϕ2(4.106)

The left side of this equation is only a function of the radius r, and the rightside of the angles ϑ and ϕ. Therefore, this equation can only be fulfilled ifeach side is equal to a constant number C, that is½

1

R

∂

∂r

µr2

∂

∂rR

¶+2m0r

2

~2

µE +

e204π ε0r

¶¾= C, (4.107)

− 1Y

∙1

sinϑ

∂

∂ϑ

µsinϑ

∂Y

∂ϑ

¶¸+

1

sin2 ϑ

∂2Y

∂ ϕ2= C. (4.108)

Spherical Harmonics

The solutions to the angular part of the Schroedinger equation (4.108)

1

sinϑ

∂

∂ϑ

µsinϑ

∂Y

∂ϑ

¶+

1

sin2 ϑ

∂2Y

∂ ϕ2+ CY = 0. (4.109)

are called spherical harmonics, see Appendix A to this chapter

Y m1 (ϑ, ϕ) = (−1)

m

s(2l + 1)

4π

(l − |m|)!(l + |m|)! P

m1 (cosϑ) e

jm ϕ

. (4.110)


In this solution, l and m are quantum numbers, whose physical meaningwe will discuss later, but from the solution, we see immediately, that mdetermines the azimutal part of the spherical harmonic and l the polar part.These numbers may take on the following values

l = 0, 1, 2, ... positive whole number, (4.111)

m = 0,±1,±2, ...,±l. (4.112)

The functions Pm1 (cosϑ) are called associated Legendre Polynomials. The

Y m1 (ϑ, ϕ) are called normalized spherical harmonics and play an importantrole whenever a partial differential equation that contains the Laplace op-erator is solved in spherical coordinates. The spherical harmonics form asystem of orthogonal functions on the full volume angle 4π, i.e. ϑ [0, π] andϕ [−π, π]

πZ0

2πZ0

Y ml∗(ϑ, ϕ)Y m0

l0 (ϑ, ϕ) sinϑ dϑ dϕ = δll0 , δmm0 . (4.113)

Therefore, a function of the angular variable (ϑ, ϕ) can be expanded in spher-ical harmonics. The spherical harmonics with negative azimuthal number -mcan be expressed in terms of those with positive azimuthal number m.

Y −m1 (ϑ, ϕ) = (−1)m (Y ml (ϑ, ϕ))

∗ . (4.114)

The lowest order spherical harmonics are listed in Table 4.1. Figure 4.10shows a cut through the spherical harmonics Y m

1 (ϑ, ϕ) along the meridionalplane.


Y00 (ϑ, ϕ)=1√4π, Y 0

1 (ϑ, ϕ) =q

34πcosϑ , Y 1

1 (ϑ, ϕ) = −q

38πsinϑ ejϕ ,

Y02 (ϑ, ϕ)=q

516π(3 cos2 ϑ− 1) , Y12 (ϑ, ϕ)=-

q158πsinϑ cosϕ ejϕ, Y

2

2 (ϑ, ϕ)=q

1532πsin2 ϑ e2jϕ,

Y03 (ϑ, ϕ)=q

716π(5 cos3 ϑ− 3 cosϑ) , Y13 (ϑ, ϕ)=-

q2164πsinϑ (5 cos2 ϑ− 1) ejϕ ,

Y23 (ϑ, ϕ)=q

10532πsin2 ϑ cosϑ ej2ϕ , Y33 (ϑ, ϕ)=-

q3564πsin3 ϑ ej3ϕ .

Table 4.1: Lowest order spherical harmonics

Figure 4.10: Lowest order spherical harmonics Y m1 (ϑ, ϕ) , along the merid-

ional plane, i.e. ϕ = 0.


Radial Hydrogen Wave Functions

From the sperical harmonics one finds, that the constant C depends only onthe polar quantum number l

C = l(l + 1), (4.115)

and the radial equation becomes

d2R

dr2+2

r

dR

d r+

µ2m0E

~2+

m0e20

2πε0~2r− l (l + 1)

r2

¶R = 0. (4.116)

The radial equation has in addition to the 1/r -Coulomb potential aquired anadditional term that can be traced back to the additional centrifugal energyof the electron, related to the angular momentum of the electron orbitingaround the nucleus.

Erot =~2

2m0

l (l + 1)

r2=

L2

2m0r2. (4.117)

Obviously, the spherical harmonics are related to the angular momentumL of the particle, because Erot is the energy of a particle due to its angu-lar momentum

¯L¯=pl (l + 1)~ and the moment of inertia m0r

2. Thusquantum mechanically, the particle can no longer access arbitrary valuesfor the angular momentum. The angular momentum can only have values¯L¯=pl (l + 1)~ with l = 0, 1, 2, ....

One finds that the radial equation (4.116) has only bound state solutions,i.e. E < 0, for discrete values of the energy equal to

E = En = −m0e

4

8ε20h2

1

n2, (4.118)

withn = 1, 2, ...and n ≥ l + 1. (4.119)

The corresponding radial wave functions take on the form

Rn1 (r) =2

n2

s(n− l − 1)![(n+ l)!]3

r−3/21 · ρ1 L21+1n−l−1 (ρ) e−ρ/2,


where r1is the Bohr radius, ρ = 2rnr1

is the normalized radius and L21+1n−l−1 (ρ)is the Laguerre polynomial [5], which can be expressed as

Lrs (x) =

sXq=0

(−1)q (s+ r)!2

(s−q) ! (r + q)!

xq

q!. (4.120)

The forefactors in the radial wave functions are chosen such that they arenormalized when weighted by the radial weight r2

∞Z0

Rnl (r) Rn0l (r) r2 dr = δn,n0 , (4.121)

The lowest order Laguerre Polynomials are summarized in Table 4.2 The

L10 (x) = 1 , L11 (x) = 4− 2x , L12 (x) = 18− 18x+ 3x2 ,

L13 (x) = 96− 144x+ 48x2 − 4x3 , L20 (x) = 2 ,

L21 (x) = 18− 6x , L22 (x) = 144− 96 + 12x2 ,

L33 (x) = 6 , L31 (x) = 96− 24x ,

L40 (x) = 24 .

Table 4.2: Lowest order Laguerre Polynomials

radial wave functions of the hydrogen atom are listed in Table 4.3 and plotsof the lowest order radial wave functions are presented in Figure 4.11


R10(r) =2√r31e−r/r1 , R20(r). =

1

2√2√

r31

³2− r

r1

´e−r/2r1

R21(r) =1

2√6√

r31

rr1e−r/2r1

R30(r) =1

81√3√

r31

µ27− 18 r

r1+ 2

³rr1

´2¶e−r/3r1

R31(r) =4

81√6√

r31

³6− r

r1

´rr1e−r/3r1 , R32(r) =

4

81√30√

r31

³rr1

´2e−r/3r1

Table 4.3: Lowest order radial wavefunctions Rn,l(r).

Figure 4.11: Radial wavefunctions Rnl(r) of the hydrogen atom.

In total we found the stationary states, or the energy eigenfunctions, of


the hydrogen atom. Those are

ψnlm (r, ϑ, ϕ) = Rnl (r) Y ml (ϑ, ϕ) . (4.122)

The lower order wave functions are listed in Table 4.4 and plots of the re-sulting probability densities of the lowest order energy eigenstates of thehydrogen atom are shown in Figure 4.12

ψ100(r, ϑ, ϕ) =1√

π√

r31e−r/r1

ψ200(r, ϑ, ϕ) =1

4√2π√

r31

³2− r

r1

´e−r/2r1

ψ210(r, ϑ, ϕ) =1

4√2π√

r31

rr1e−r/2r1 cosϑ

ψ21±1(r, ϑ, ϕ) =1

8√π√

r31

rr1e−r/2r1 sinϑe±jϕ,

ψ300(r, ϑ, ϕ) =1

81√3π√

r31

µ27− 18 r

r1+ 2

³rr1

´2¶e−r/3r1

Table 4.4: Lowest order hydrogen wavefunctions ψn,l,m(r, ϑ, ϕ).

For each value of l, there are 2l + 1 possible values for the azimutalquantum number m. These are −l,−(l − 1), ....,−1, 0, 1, ..., l − 1, l. As aresult, there are n2 different stationary states for each value of n. Forhistoric reasons the states with l = 0 are called s-states (sharp), the stateswith l = 1 are called p-states (principle), the states with l = 2 are calledd-states (diffuse) and the states with l = 3 are called f-states (fundamental).Figure 4.13 shows the surfaces of equal propability density for the first threeexcited states (n=2) of the hydrogen atom in 3 dimensions.


Figure 4.12: Probability densities of the lowest order hydrogen wavefunctions.(The density is presented along the meridial plane).


ψ310(r, ϑ, ϕ) =1

81√π√

r31

³6− r

r1

´e−r/3r1 cosϑ

ψ31±1(r, ϑ, ϕ) =1

81√π√

r31

³6− r

r1

´rr1e−r/3r1 sinϑe±jϕ

ψ320(r, ϑ, ϕ) =1

81√6π√

r31

r2

r21e−r/3r1 (3 cos2 ϑ− 1)

ψ32±1(r, ϑ, ϕ) =1

81√π√

r31

r2

r21e−r/3r1 sinϑ cosϑe±jϕ,

ψ32±2(r, ϑ, ϕ) =1

162√3π√

r31

r2

r21e−r/3r1 sin2 e±2jϕ

Table 4.5: Lowest order hydrogen wavefunctions ψn,l,m(r, ϑ, ϕ).continued.

x y

z2p(m=+/-1)

x y

z2s

x y

z2p(m=0)

Figure 4.13: Surfaces of constant propability density for the first three excitedstates (n=2) of the hydrogen atom.

4.5.3 Energy Spectrum of Hydrogen

We haven’t yet discussed the energy eigenspectrum of hydrogen followingfrom the solution of the stationary Schroedinger Equation. From Eqs.(??)and (3.87) we find this to be

E = − m0e4

8 ε20h2

1

n2, (4.123)


which also agrees with the energy spectrum of the Bohr-Sommerfeld model,see section 3.5. The lowest energy eigenstate is

E1 = −m0e

4

8 ε20h2= −13.7eV. (4.124)

The energy eigenvalues constitute a sequence that converges for large n→∞towards 0, which corresponds to removing the electron from the atom. Theenergy to do so is E∞ −E1 = 13.7eV.

Figure 4.15 shows the energy levels and the term diagram of the hydrogenatom and how the Lyman, Balmer, Paschen, Brackett and Pfund series arisefrom it. Each wavefunction is uniquely described by the set of quantumnumbers (n,l,m). The first quantum number n specifies the energy eigenvalue En. As we will show in problem sets, the second quantum numberl determines the eigenvalue of the squared angular momentum operator L2

with eigenvalues

L2 ψnlm (r, ϑ, ϕ) = l(l + 1)~2 ψnlm (r, ϑ, ϕ) , (4.125)

which determines the rotational energy of the corresponding stationary state.And the third quantum number m detemines the eigenvalue of the operatordescribing the z-component of the angular momentum operator

Lz ψnlm (r, ϑ, ϕ) = m~ ψnlm (r, ϑ, ϕ) . (4.126)

The 2l + 1 different values for m, describe the possible values for the z-component of the angluar momentum for a given value of l. Figure 4.14.shows the possible values for the z-component of the angular moment vectorfor l = 1 and l = 2.


Figure 4.14: Possible values for the z-component of the angular momentumvector for the cases l = 1 and l = 2.

Figure 4.15: Energy levels and term diagram for the hydrogen atom [3]

In fact, the description of the electron wave functions is not yet complete,because the electron has an internal degree of freedom, that is its spin. Thespin is an internal angular momentum of the electron that carries a magneticmoment with it. The Stern-Gerlach experiment shows that this degree of


freedom has two eigenstates, i.e. the spin can be oriented parallel or anti-parallel to the direction of an applied magnetic field. The values of theinternal angluar mometum with respect to the quantization axis defined byan external field, that shall be chosen along the z-axis, are s = ±~/2. Thusthe energy eigenstates of an electron in hydrogen are uniquely characterizedby four quantum numbers, n, l, m, and s. As Figure 4.15 shows, the energy

spectrum is degenerate, i.e. for n > 1, there exist to each energy eigenvalueseveral eigenfunctions, that are only uniquely characterized by the additionalquantum numbers for angular momentum and spin. This is called degeneracybecause there exist to a given energy eigenvalue several physically differenteigen states.

4.5.4 Superposition States, Radiative Transitions andSelection Rules

If the atom is in one of its energy eigenstates it is obvious by symmetry thatthe average position of the electron calculated via

hri =Z ∞

−∞r |Ψ (r, t)|2 d3r = 0 (4.127)

vanishes. Then also the average dipole moment p = −e hri vanishes and,therfore, the atom does not radiate in a stationary state as postulated inthe Bohr model. Here, it is a natural outcome. No postulate is needed.However, when the atom is in a superposition state, for example betweenground state and the first excites states, the average position of the electronor dipole moment of the atom does not any longer vanish and the atom hasa dipolement that radiates like a classical dipole does. For example Figure4.16 shows the charge distribution as a function of time for an atom in thesuperposition state between the 1s-groundstate and the 2p(m = 0) excitestate

1√2

¡ψ1s(r, t) + ψ2p,m=0(r, t)

¢=

1√2(ψ100(r, t) + ψ210(r, t)) = . (4.128)

=1√

2πpr31

µe−r/r1e−jE1t/~ +

1

4√2

r

r1e−r/2r1 cosϑe−jE2t/~

¶.

In the propability density, i.e. the magnitude square of the wave function thecontributions between the ground state and excited state interfer positively


or negatively depending on the relative phase between the two wavefunctions,which depends on the phase angle ∆Et/~, with ∆E = E2 −E1..

Figure 4.16: Charge distribution in the superposition state1√2

¡ψ1s(r, t) + ψ2p,m=0(r, t)

¢for different times. The atom have a dipole

moment in the z-direction [6].

Figure 4.16 shows the charge distribution as a function of time for an atomin the superposition state between the 1s-groundstate and the 2p(m = 1)excite state. Instead of an oscillating charge distribution, the atom showsnow a rotating dipole, which emittes a circular polarized electromagneticwave.

1√2

¡ψ1s(r, t) + ψ2p,m=1(r, t)

¢=

1√2(ψ100(r, t) + ψ211(r, t)) . (4.129)

=1√

2πpr31

µe−r/r1e−jE1t/~ +

1

8

r

r1e−r/2r1 sinϑejϕe−jE2t/~

¶.

Figure 4.17: Charge distribution in the superposition state1√2

¡ψ1s(r, t) + ψ2p,m=1(r, t)

¢for different times. The atoms show a

dipole moment rotating in the x-y plane [6].


Figure 4.18 summarizes the polarization of the electromagnetic field emit-ted from a superposition state with an oscillating dipole in the z-direction(m = 0) (a) and a rotating dipol in the x-y-plane (m = 1)(b). The transitionwith (m = −1) emittes polarized light with the opposite circularlity then thetransition with (m = 1).

Figure 4.18: Polarization of the associated electromagnetic radiation emittedduring the transition of the atom for (a) an oscillating dipole and (b) from arotating dipole [6].

If an atomic gas in termal equilibrium no orientation of the atom is pref-ered and it emittes light from all possible transitions equally and, therefore,the light is not polarized. However, if the atom is put into a magnetic field,which is oriented in the z-direction, the energy levels for the different p-


states split depending on the magnetic quantum number m. This effect iscalled Zeeman effect. Because, with the orbital angular momentum L of theelectron state there is an electrical current associated with it, that leads toa magnetic moment of that particular state and with it to en energy shift ofthe corresponding energy level, see Figure 4.19. Then the light emitted atthe different frequencies is polarized correspondingly. Or light with a givenpolarization interacts only with the corresponding transition, where it caninduce the corresponding dipolemoment in the atom.

Figure 4.19: Zeeman effect and related energy levels for the m=0 and m=+/-1 transitions.

4.6. WAVE MECHANICS 233

As we will see in more detail later, the interaction strength between lightand an atom via dipole transistions between energy eigenstates ψa and ψb isdetermined by the transition matrix element

Mab = e0

Z ∞

−∞ψ∗a (r) r ψb (r) d3r (4.130)

As we have seen for the possible dipole transitions between states with prin-ciple quantum number n=1 and n=2, the transistions are related to a changein orbital quantum number

∆l = ±1 and m = 0,±1.

It turns out that this is in generel the selection rule for dipole radiation.Finally, we schould mention, that also higher order transitions are possible,such as quadrupole transitions. However, at optical frequences these transi-tions are much weaker than dipole transitions.

4.6 Wave Mechanics

In this section, we generalize the concepts we have learned in the previoussections. The goal here is to give a broader description of quantummechanicsin terms of wave functions that are solutions to the Schroedinger Equation.In classical mechanics the particle state is determined by its position

and momentum and the state evolution is determined by Newton’s law. Inquantum mechanics the particle state is completely described by its wavefunction and the state evolution is determined by the Schroedinger equation.The wave function as a complete description of the particle enables us to

compute expected values of physical quantities of the particle when a cor-responding measurement is performed. The measurement results are realnumbers, like the energy, or position or momentum the particle has in thisstate. The physically measureable quantities are called observables. In clas-sical mechanics these observables or real variables like x for position, p formomentum or functions thereof, like the energy, which is called the Hamil-tonian H(p, x) = p2

2m+ V (x) in classical mechanics. For simplicity, we state

the results only for one-dimensional systems but it is straight forward to ex-tend these results to multi-dimensional sytems. In quantum mechanics these


observables become operators:

x : position operator (4.131)

p =~j∂

∂x: momentum operator (4.132)

H(p, x) = − ~2

2m

∂2

∂x2+ V (x) : Hamiltonian operator (4.133)

If we carry out measurements of these observables, the result is a real numberin each measurement and after many measurements on identical systems wecan make a statistics of these measurements and the statistics is completelydescribed by the moments of the observable.

4.6.1 Position Statistics

The statistical interpretation of quantum mechanics enables us to computethe expected value of the position operator or any of its moments accordingto

hxi =

Z ∞

−∞Ψ∗ (x, t) x Ψ (x, t) dx (4.134)

hxmi =

Z ∞

−∞Ψ∗ (x, t) xm Ψ (x, t) dx (4.135)

The expectation value of functions of operators can always be evaluated bydefining the operator by its Taylor expansion

hf(x)i =

Z ∞

−∞Ψ∗ (x, t) f(x) Ψ (x, t) dx (4.136)

=

* ∞Xn=0

1

n!f (n)(0) xn

+

=∞Xn=0

1

n!f (n)(0)

¿Z ∞

−∞Ψ∗ (x, t) xn Ψ (x, t) dx

À

4.6.2 Momentum Statistics

The momentum statistics is then


hpi =Z ∞

−∞Ψ∗ (x, t)

~j∂

∂xΨ (x, t) dx (4.137)

which can be written in terms of the wave function in the wave number space,which we define now for symmetry reasons as the Fourier transform of thewave function where the 2π is symmetrically distributed between Fourier andinverse Fourier transform

φ (k, t) =1√2π

Z ∞

−∞Ψ (x, t) e−jkx dx, (4.138)

Ψ (x, t) =1√2π

Z ∞

−∞φ (k, t) ejkx dk. (4.139)

Using the differentiation theorem of the Fourier transform and the generalizedParseval relationZ ∞

−∞φ∗1 (k) φ2 (k) dk =

Z ∞

−∞Ψ∗1 (x)Ψ2 (x) dx (4.140)

we find

hpi =

Z ∞

−∞φ∗ (k, t) ~k φ (k, t) dk (4.141)

=

Z ∞

−∞~k |φ (k, t)|2 dk. (4.142)

The introduction of the symmetrically defined expectation value of an oper-ator according Eq.(4.134), where x can stand for any operator can be carriedout using the wave function in the position space or the wave number spaceusing the corresponding represenation of the wave function and of the oper-ator.

4.6.3 Energy Statistics

The analysis for the measurement of position or moment carries over to everyobservable in an analogous way. Thus the expectation value of the energy is

hH(x, p)i =

Z ∞

−∞Ψ∗ (x, t) H(x, p) Ψ (x, t) dx (4.143)

=

Z ∞

−∞Ψ∗ (x, t)

µ− 1

2m~2∂2

∂x2+ V (x)

¶Ψ (x, t) dx.(4.144)


If the system is in an energy eigenstate, i.e.

Ψ (x, t) = ψn (x) ejωnt (4.145)

withH(x, p) ψn (x) = En ψn (x) , (4.146)

we obtain

hH(x, p)i =Z ∞

−∞Ψ∗ (x, t) En Ψ (x, t) dx = En. (4.147)

If the system is in a superposition of energy eigenstates

Ψ (x, t) =∞Xn=0

cnψn(x)ejωnt. (4.148)

we obtain

hH(x, p)i =∞Xn=0

En |cn|2 . (4.149)

4.6.4 Arbitrary Observable

There may also occur observables that are not simple to translate from theclassical to the quantum domain, such as the product

pcl · xcl = xcl · pcl (4.150)

Classically it does not matter which variable comes first. However, if wetranfer this expression into quantum mechanics, the corresponding operatordepends on the odering, for example

pqm · xqmΨ (x, t) =~j∂

∂x(xΨ (x, t)) = (4.151)

=~jΨ (x, t) +

~jx∂

∂xΨ (x, t) , (4.152)

=

µ~j+ xqm · pqm

¶Ψ (x, t) . (4.153)

The decision of which expression represents the correct quantum mechanicaloperator or eventually even a linear combination of the possible expressions,has to be based on a close examination of the actual measurement apparatus


that would measure the corresponding observable. Finally, the expressionalso has to deliver results that are in agreement with experimental findings.If we have an operator that is a function of x and p and we have decided

on a unique expression in terms of a power expansion in x and p

g(x, p)→ gop(x,~j∂

∂x) (4.154)

then we can compute its expected value either in the space domain or thewave number domain

hgopi =

Z ∞

−∞Ψ∗ (x, t) gop(x,

~j∂

∂x) Ψ (x, t) dx (4.155)

=

Z ∞

−∞φ∗ (k, t) gop(j

∂

∂k, ~k) φ (k, t) dk (4.156)

That is this operator can be represented either in real space or in k-space asgop(x,

~j∂∂x) or gop(j ∂

∂k, ~k).

4.6.5 Eigenfunctions and Eigenvalues of Operators

A differential operator has in general eigenfunctions and corresponding eigen-values

gop(x,~j∂

∂x) ψn (x) = gnψn (x) , (4.157)

where gn is the eigenvalue to the eigenfunction ψn (x) . An example for adifferential operator is the Hamiltonian operator describing a partical movingin a potential

Hop = −1

2m~2∂2

∂x2+ V (x) (4.158)

the corresponding eigenvalue equation is the stationary Schroedinger Equa-tion

Hopψn (x) = Enψn (x) . (4.159)

Thus the energy levels of a quantum system are the eigenvalues of the corre-sponding Hamiltonian operator.The operator for which


Zψ∗n (x) (Hopψm (x)) dx =

Z(Hopψn (x))

∗ ψm (x) dx, (4.160)

for arbitrary wave functions ψn and ψm is called a hermitian operation. Fromthis equation we find immediately that the expected values of a hermitianoperator are real, which also has the consequence that the eigenvalues ofhermitian operators are real. This is important since operators that representobservables must have real expected values and real eigenvalues since theseare results of physical measurements, which are real. Thus observables arerepresented by hermitian operators. This is easy to proove. Let’s assume wehave found two eigenfunctions and the corresponding eigen values

gopψm = gmψm, (4.161)

gopψn = gnψn. (4.162)

Then Zψ∗ngopψm dx = gm

Zψ∗nψm. (4.163)

By taking advantage of the fact that the operator is hermitian we can alsowrite Z

ψ∗ngopψm dx =

Z(gopψn)

∗ ψm dx = g∗n

Zψ∗nψm dx (4.164)

The right sides of Eqs.(4.163) and (4.164) must be equal

(gm − g∗n)

Zψ∗nψm dx = 0 (4.165)

If n = m the integral can not vanish and Eq.(4.165) enforces gn = g∗n, i.e.the corresponding eigenvalues are real. If n 6= m and the correspondingeigenvalues are not degenerate, i.e. different eigenfunctions have differenteigenvalues, then Eq.(4.165) enforces that the eigenfunctions are orthogonalto each other Z

ψ∗nψm dx = 0, for n 6= m. (4.166)

Thus, if there is no degneracy, the eigenfunctions of a hermitian operator areorthogonal to each other. If there is degeneracy, one can always choose an or-thogonal set of eigenfunctions. If the eigenfunctions are properly normalizedRψ∗nψn dx = 1, then the eigenfunctions build an orthonormal systemZ

ψ∗nψm dx = δnm, (4.167)


and are complete, i.e. any arbitrary function f (x) can be expressed as asuperposition of the orthonormal basis functions ψn (x)

f (x) =∞Xn=0

cnψn (x) . (4.168)

Thus we can freely change the basis in which we describe a certain physicalproblem. To account fully for this fact, we no longer wish to use wave me-chanics, ie. express the wave function as a function in position space or ink-space. Instead we will utilize a vector in an abstract function space, i.e. aHilbert space. In this way, we can formulate a physical problem, without us-ing a fixed representation for the state of the system (wave function) and thecorresponding operator representations. This description enables us to makefull use of the mathematical structure of Hilbert spaces and the algebraicproperties of operators.

4.6.6 Appendix A: Spherical Harmonics

In this appendix, we present a brief derivation of the spherical harmonics,which are a solution of the equation

1

sinϑ

∂

∂ϑ

µsinϑ

∂Y

∂ϑ

¶+

1

sin2 ϑ

∂2Y

∂ ϕ2+ CY = 0. (4.169)

It is obvious, that after multiplication with sin2 ϑ the equation can be sepa-rted again in two equations for the polar and azimutal functions

Y (ϑ, ϕ) = θ(ϑ) · φ (ϕ) (4.170)

withd2φ

dϕ2+m2φ = 0 , (4.171)

1

sinϑ

d

dϑ

µsinϑ

dθ

dϑ

¶+

µC − m2

sin2 ϑ

¶θ = 0. (4.172)

Since the wavefunction has to be periodic in the azimutal angle ϕ, with period2π, m must be a whole number and the solutions are either sines or cosinesor the complex solutions

φ (ϕ) = const. ejmϕ, with m = . . .− 2,−1, 0, 1, 2 . . . . (4.173)


Now, we are left to solve the polar equation (4.172), which can be transformedinto Legendre’s differential equation with the substitution

ξ = cosϑ (4.174)

into the Legendre’s differential equation, well known from electrostatics withspherical boundary conditions.

¡1− ξ2

¢ d2θdξ2− 2ξ dθ

dξ+

µC− m2

1− ξ2

¶θ = 0 . (4.175)

It turns out, that this equation has only bounded solutions on the intervalξ [−1, 1], if the constant C is a whole number of the form

C = l (l + 1) with l = 0, 1, 2 . . . . (4.176)

For a given number l, there are only 2l + 1 different solutions for m

m = −l,−l + 1, . . . − 1, 0, 1 . . . l − 1, l (4.177)

For m = 0, Eq.(4.175) is the regular Legendre’s differential equation and thesolutions are the Legendre-Polynomials [5]

P0 (ξ) = 1 , P3 (ξ) =52ξ3 − 3

2ξ ,

P1 (ξ) = ξ , P4 (ξ) =358ξ4 − 15

4ξ2 + 3

8,

P2 (ξ) =32ξ2 − 1

2, P5 (ξ) =

638ξ5 − 35

4ξ3 + 15

8ξ .

(4.178)

For m 6= 0, Eq.(4.175) is the associated Legendre’s differential equation andthe solutions are the associated Legendre-Polynomials, which can be gener-ated from the Legendre-Polynomials by

Pm1 (ξ) =

¡1− ξ2

¢m/2 dmP1 (ξ)

dξm. (4.179)

Overall the angular functions can be combined to form the spherical harmon-ics

Y m1 (ϑ, ϕ) = (−1)

m

s(2l + 1)

4π

(l − |m|)!(l + |m|)! P

m1 (cosϑ) e

jm ϕ

. (4.180)


The fore factor is chosen such that they are normalized over the full volumeangle 4π, i.e. ϑ [0, π] and ϕ [−π, π] taking the polar weighting factor sinϑinto account

πZ0

2πZ0

Y ml∗(ϑ, ϕ)Y m0

l0 (ϑ, ϕ) sinϑ dϑ dϕ = δll0 , δmm0 . (4.181)

The spherical harmonics form a system of orthonormal functions on the fullvolume angle. Any function of the angular variables (ϑ, ϕ) can be expandedin spherical harmonics. The spherical harmonics with negative azimuthalnumber -m can be expressed in terms of those with positive azimuthal numberm.

Y −m1 (ϑ, ϕ) = (−1)m (Y ml (ϑ, ϕ))

∗ . (4.182)

4.6.7 Appendix B: Radial Wave Function

After choosing the spherical harmonic with indices l,m the radial Equation(4.107) becomes

d2R

dr2+2

r

dR

d r+

µ2m0E

~2+

m0e20

2πε0~2r− l (l + 1)

r2

¶R = 0. (4.183)

It is advantageous to introduce normalized quantities for the energy and theradial variable

E =E1n2

(4.184)

ρ =2r

nr1, (4.185)

which transforms the radial equation into the dimensionless equation

d2R

dρ2+2

ρ

dR

dρ+

µ−14+n

ρ− l (l + 1)

ρ2

¶R = 0. (4.186)

We first consider who the solution to this equation has to behave asymptot-ically for ρ→∞. For large ρ, we end up with the equation

d2R

dρ2− 14R = 0, (4.187)


which has only one useful solution to construct a bound state solution, thatis e−ρ/2. Therefore, it makes sense to look for solutions to the full radialequation (4.186) of the form

R(ρ) = ρsw(ρ) e−ρ/2. (4.188)

This trial solution leads to

ρ2d2w

dρ2+ ρ [2 (s+ 1)− ρ]

dw

d ρ+ [ρ(n− s− 1) + s(s+ 1)− l(l + 1)] w = 0.

(4.189)This equation only works for ρ = 0, if s = l or s = −(l + 1). The latterchoice is not possible, because otherwise the radial solution diverges at theorigin. We are then left over to solve the equation

ρ2d2w

dρ2+ ρ [2 (l + 1)− ρ]

dw

dρ+ [ρ(n− l − 1)] w = 0. (4.190)

We try a polynomial solution

w(ρ) = b0 + b1ρ+ b2ρ2 + .... (4.191)

and after substitution into Eq.(4.191) and comparing coefficients we find therecursive relation

bk+1 =k + l + 1− n

(k + 1)(k + 2l + 2)bk. (4.192)

Forn = p+ l + 1 (4.193)

the series stops after the term with index p. If this is not the case, theseries does not stop and the coefficients asymptotically approach bk+1 = bk/kfor k → ∞. But this would indicate that w(ρ)˜eρ for large ρ, which wouldlead to an unbounded wave function. Therefore, n must be a positive wholenumber and because of condition (4.193)

n ≥ l + 1.

It turns out that the function the differential equation (4.190) is related toLaguerre’s differential equation [5] and its solutions can be expressed in terms


of the Laguerre polynomial L21+1n−l+1 (ρ) , such that the radial wave functionscan be written as

Fnl (ρ) = ρ1 L21+1n−l−1 (ρ) e−ρ/2. (4.194)

The Laguerre polynamials can be expressed as

Lrs (x) =

sXq=0

(−1)q (s+ r)!2

(s−q) ! (r + q)!

xq

q!. (4.195)

The radial functions again form an orthogonal system of functions

∞Z0

Fnl (ρ)Fn0l (ρ) ρ2dρ =

2n [(n+ l)!]3

(n− l − 1)! δnn0 . (4.196)

After substituting back the normalization of the radial coordinate fromEq.(4.184),we find the radial wave function to be

Rn1 (r) = Nnl Fnl (ρ) . (4.197)

The normalization factor is determined by

∞Z0

Rnl (r) Rn0l (r) r2 dr = δn,n0 , (4.198)

which gives

Nnl =2

n2

s(n− l − 1)![(n+ l)!]3

r−3/21 . (4.199)

Bibliography

[1] Introduction to Quantum Mechanics, Griffiths, David J., Prentice Hall,1995


[3] The Physics of Atoms and Quanta, Haken and Wolf, Springer Verlag1994.

[4] Practical Quantum Mechanics, S. Flügge, Springer Verlag, Berlin, 1999.

[5] Handbook of Mathematical Functions, Abramowitz and Stegun, DoverPublications, NY 1970.

[6] Introduction to Modern Optics, G. R. Fowles, Dover Publications, NY1989.

245

246 BIBLIOGRAPHY

Chapter 5

Interaction of Light and Matter

Usually, there are infinitely many energy eigenstates in an atomic, molecularor solid-state medium and the spectral lines of the emission are associatedwith transitions between two of these energy eigenstates.As we have seenform the treatment of the hydrogen atom, atomic and also molecular gasesin low concentration show discrete energy eigen spectra. For many physi-cal considerations it is already sufficient to take only two of these possibleenergy eigenstates into account, for example those which are related to thelaser transition. The pumping of the laser can be later introduced by phe-nomenological relaxation processes that describe the pumping of the laservia other levels. The resulting simple model is often called a two-level atom,which is mathematically also equivalent to a spin 1/2 particle in an externalmagnetic field, because the spin can only be parallel or anti-parallel to thefield, i.e. it has two energy levels and energy eigenstates [4]. The interactionof the two-level atom with the electric field of an electromagnetic wave isdescribed by the Bloch equations, which where originally used to describenuclear magnetic resonance phenomena.

5.1 The Two-Level Model

An atom with only two energy eigenvalues is described by a two-dimensionalstate space spanned by the two energy eigenstates with wave functions ψg(r)and ψe(r). The two states constitute a complete orthonormal system. Thecorresponding energy eigenvalues are Eg and Ee for ground and excited state,see Fig. 5.1. Fig. 5.1 shows a one-dimensional potential of finite depth that

247

248 CHAPTER 5. INTERACTION OF LIGHT AND MATTER

Figure 5.1: One dimensional model for a two-level atom.

has only two bound energy eigenstates, which can serve as a model for thefollowing discussion. The Hamiltonian operator of the two-level atom shellbe denoted HA, with

HA ψe(r) = Ee ψe(r) (5.1)

HA ψg(r) = Eg ψg(r). (5.2)

An arbitrary state is a superposition state of ground and excited state

Ψ(r, t) = cg(t) ψg(r) + ce(t) ψe(r). (5.3)

The coefficients cg(t) and ce(t) are the propability amplitudes to find theatom in the ground or excited state, respectively:

|cg|2 : propability to find the atom in the ground state (5.4)

|ce|2 : propability to find the atom in the excited state (5.5)

In this two-dimensional state space the temporal dynamics is described bythe time dependence of the coefficients. The time dependence follows fromthe Schroedinger Equation:

j ~ ∂∂tΨ (r, t) = HA Ψ (r, t) (5.6)

j ~¡cg(t) ψg(r) + ce(t) ψe(r)

¢=

¡Eg cg(t) ψg(r) +Ee ce(t) ψe(r)

¢.(5.7)

5.2. THEATOM-FIELD INTERACTION INDIPOLEAPPROXIMATION249

By multiplication of this equation from the left with the complexe conjugateground state or the excited state and integration over x using the orthogo-nality relations for the energy eigenstates, we obtain to separate equationsfor the time dependence of the coefficients:

ce = −jωece, with ωe = Ee /~, (5.8)

cg = −jωgcg, with ωg = Eg /~. (5.9)

As we will discuss in more detail later, this procedure is equivalent to pro-jecting the Schroedinger Equation onto the energy eigenstates. The timedependent solution of the Schroedinger Equation of the free atom with ini-tial conditions cg(0) and ce(0) is then

Ψ(r, t) = cg(0)e−jωgt ψg(r) + ce(0)e

−jωet ψe(r). (5.10)

The question that arises now is how does the atomic dynamics change in thepresence of an external electro-magnetic field and environmental perturba-tions?

5.2 The Atom-Field Interaction In Dipole Ap-proximation

The strongest interaction between atoms or molecules and electromagneticfields is due to the induced dipole moment in the atom or molecule in thepresence of an electric field. The dipole moment of an atom d is determinedby the position d of the electron from the nucleus via

d = −e0r. (5.11)

A dipole in an elecric fieldE has the additional energyHd = −d·E. Therefore,we may argue that the Schroedinger Equation for an atom in an electromagnetic field taking the dipole interaction into account may be describedby the total hamiltonian operator

HAF = HA − d · E(rA, t), (5.12)

where rA is the position of the atom and E(rA, t) = E(t) the field at theposition of the atom. The new Schroedinger Equation using the hamiltonian


(5.12) leads to new equations of motion for the propability amplitudes whenprojected onto the corresponding energy eigenstates, as done before to deriveeqs. (5.8) and (5.9). During these projections the following matrix elementsof the dipolmoment of the atom arise

Mee =

Zψ∗e(r) d ψe(r) dr = −e0

Zψ∗e(r) r ψe(r), (5.13)

Meg =

Zψ∗e(r) d ψg(r) dr = −e0

Zψ∗e(r) r ψg(r), (5.14)

Mge =

Zψ∗g(r) d ψe(r) dr =M∗

∈g, (5.15)

Mgg =

Zψ∗g(r) d ψg(r) dr = −e0

Zψ∗g(r) r ψg(r). (5.16)

As we have seen when discussing the hydrogen atom in the last section,due to the symmetry of atomic energy eigenstates, the matrix dipole elementsRψ∗e(r) d ψe(r) dr and

Rψ∗g(r) d ψg(r) dr, vanish, i.e. there is no atomic

dipolmoment if the atom is in an energy eigenstate. This might not be thecase for energy eigenstates in a solid. The atoms consituting the solid are ori-ented in a lattice, which may break the symmetry. If so, there are permanentdipole moments and consequently the matrix elements

Rψ∗e(r) d ψe(r) dr andR

ψ∗g(r) d ψg(r) dr would not vanish. Here, we assume atomic wave func-tions, i.e. symmetric or anti-symmetric wave functions and therefore the newequations of motion for the propability amplitudes become

ce = −jωece + jcg1

~

µZψ∗e(r) d ψg(r) dr

¶·E(t), (5.17)

cg = −jωgcg + jce1

~

µZψ∗g(r) d ψe(r) dr

¶· E(t). (5.18)

Separating the electric field into its polarization vector e and excitation am-plitude

E(t) = E(t) e, (5.19)

the Schroedinger Equation becomes

ce = −jωece + jcgMeg · e~

E(t), (5.20)

cg = −jωgcg + jceM∗

eg · e~

E(t). (5.21)

5.2. THEATOM-FIELD INTERACTION INDIPOLEAPPROXIMATION251

The expected value for the dipole moment of an atom in state (5.3) can alsobe expressed in terms of the dipole matrix elementsD

dE= |ce|2Mee + |cg|2Mgg + c∗ecgMeg + c∗gceMge

= c∗ecgMeg + c.c., (5.22)

where, the last line again follows from the symmetry of the wavefunctions.The abreviation c.c. stands for the complex conjugate expression to the ex-pression immediatly preceding. This equation shows again that the atomposesses only a dipole moment if it is in a superposition of energy eigenstates.The Schroedinger equation, expressed as Eqs. (5.20) and (5.21), indicatesthat an external eletric field is capable of coupling two energy eigenstates,i.e. induces a dipole moment in an atom. To obtain more insight into thedynamics induced by the electric field in the atom, we look at the case of amonochromatic electric field at the position of the atom

E(t) =1

2

¡E0e

jωt +E∗0e−jωt ¢ , (5.23)

where E0 is the complex electric field amplitude. We expect strong inter-action between the field and the atom if the atomic transition frequencybetween the states, ωeg = ωe − ωg, is close to the frequency of the driv-ing field, i.e. ωeg ≈ ω. It is advantageous to transform to new probabilityamplitudes, that take some trivial oscillations already into account

Ce = ce ej(ωe+ωg+ω2

t) (5.24)

Cg = cg ej(ωe+ωg−ω2

t) (5.25)

which leads to the new equations of motion

Ce =

∙j

µωe + ωg + ω

2

¶− jωe

¸cee

j(ωe+ωg+ω2t) + jcg

Meg · e~

E(t) ej(ωe+ωg+ω

2t),

Cg =

∙j

µωe + ωg − ω

2

¶− jωg

¸cge

j(ωe+ωg−ω2t) + jce

M∗eg · e~

E(t) ej(ωe+ωg−ω

2t).

Introducing the detuning between the atomic transistion and the electric fieldfrequencies

∆ =ωeg − ω

2(5.26)


and the Rabi-frequency

Ωr =M∗

eg · e~

¡E0 +E∗0e

−j2ωt¢ , (5.27)

we obtain the following coupled mode equations for the probability ampli-tudes.

d

dtCe = −j∆Ce + j

Ω∗r2Cg, (5.28)

d

dtCg = +j∆Cg + j

Ωr

2Ce. (5.29)

If the Rabi-frequency is small |Ωr| << ωeg ≈ ω, the Rotating-Wave Ap-proximation (RWA) [3], can be used, where we only keep the slowly varyingcomponents in the interaction, i.e.

Ωr ≈M∗

eg · e~

E0 = const.. (5.30)

5.3 Rabi-Oscillations

Note, equations (5.28) and (5.29) are identical to the coupled mode equa-tions between two waveguide modes as studied in section 2.6.4. But now thecoupling is between modes in time, i.e. resonances. The modes are electronicones instead of photonic modes. But otherwise what has been said in sec-tion 2.6.4 applies in the same way. For example conservation of power flowbecomes now conservation of probability. For the case of vanishing detuningthe exchange in probability, excitation, between the modes is the strongest.For that case it is especially easy to eliminate one of the variables and wearrive at

d2

dt2Ce = − |Ωr|2

4Ce (5.31)

d2

dt2Cg = − |Ωr|2

4Cg. (5.32)

The solution to this set of equations are oscillations. If the atom is at timet = 0 in the ground-state, i.e. Cg(0) = 1 and Ce(0) = 0, respectively, we

5.3. RABI-OSCILLATIONS 253

arrive at

Cg(t) = cos

µ|Ωr|2

t

¶(5.33)

Ce(t) = −j sinµ|Ωr|2

t

¶. (5.34)

Then, the probabilities for finding the atom in the ground or excited stateare

|cg(t)|2 = cos2µ|Ωr|2

t

¶(5.35)

|ce(t)|2 = sin2µ|Ωr|2

t

¶, (5.36)

as shown in Fig. 5.2. For the expectation value of the dipole operator underthe assumption of a real dipole matrix element Meg =M∗

eg we obtainDdE= Megcec

∗g + c.c. (5.37)

= −Meg sin (|Ωr| t) sin (ωegt) . (5.38)

The coherent external field drives the population of the atomic system be-tween the two available states with a period Tr = 2π/Ωr. Applying the fieldonly over half of this period leads to a complete inversion of the population.These Rabi-oscillations have been observed in various systems ranging fromgases to semiconductors. Interestingly, the light emitted from the coherentlydriven two-level atom is not identical in frequency to the driving field. Ifwe look at the Fourier spectrum of the polarization according to Eq.(5.38),we obtain lines at frequencies ω± = ωeg ± Ωr. This is clearly a nonlinearoutput and the sidebands are called Mollow-sidebands [2]. Most importantfor the existence of these oscillations is the coherence of the atomic systemover at least one Rabi-oscillation. If this coherence is destroyed fast enough,the Rabi-oscillations cannot happen and it is then impossible to generate in-version in a two-level system by interaction with light. This is the case for alarge class of situations in light-matter interaction and especially for typicallaser materials. So we are interested in finding out what happens in the caseof loss of coherence in the atomic system due to additional interaction of theatoms with its environment.


Figure 5.2: Evolution of occupation probabilities of ground and excited stateand the average dipole moment of a two-level atom in resonant interactionwith a coherent classical field.

The coherent external field drives the population of the atomic systembetween the two available states with a period Tr = 2π/Ωr. Applying the fieldonly over half of this period leads to a complete inversion of the population.The population inversion is defined as

w = Pe − Pg = |ce|2 − |cg|2 (5.39)

These Rabi-oscillations have been observed in various systems rangingfrom gases to semiconductors. Interestingly, the light emitted from the co-herently driven two-level atom is not identical in frequency to the drivingfield. If we look at the Fourier spectrum of the polarization according toEq.(5.38), we obtain lines at frequencies ω± = ωeg ± |Ωr| . This is clearly anonlinear output and the sidebands are called Mollow-sidebands [2]. Mostimportant for the existence of these oscillations is the coherence of the atomicsystem over at least one Rabi-oscillation. If this coherence is destroyed beforesignificant population between the levels is exchanged, the Rabi-oscillationscannot happen and it is then impossible to generate inversion in a two-levelsystem by interaction with light. This is the case for a large class of situations

5.3. RABI-OSCILLATIONS 255

in light-matter interaction and especially for typical laser materials. So weare interested in finding out what happens in the case of loss of coherence inthe atomic system due to additional interaction of the atoms with its environ-ment. These non energy preserving processes can not be easily included inthe Schroedinger Equation formalism. However, we can treat these processesphenomenologically in the equations of motion for the expectation values ofthe dipol moment and the population inversion, which are of interest becausethose qunatities feed back into Maxwell’s Equations as a driving term.From the equations of motion for the coefficients of the wave function,

Eqs. (5.28) and (5.29), we derive equations of motion for the complex slowlyvarying dipole moment defined as

d = c∗ecge−jωt = C∗eCg. (5.40)

and by applying the product rule we find

d

dtd =

µd

dtC∗e

¶Cg + C∗e

µd

dtCg

¶(5.41)

= j∆C∗eCg − jΩr

2C∗gCg + j∆C∗eCg + j

Ωr

2C∗eCe (5.42)

= j∆d+ jΩr

2· w (5.43)

and

d

dtw =

µd

dtCe

¶C∗e .−

µd

dtCg

¶C∗g + c.c. (5.44)

=

µ−j∆CeC

∗e + j

Ω∗r2CgC

∗e − j∆CgC

∗g − j

Ωr

2CeC

∗g

¶+ c.c. (5.45)

= +jΩ∗rd+ c.c (5.46)

For the monochromatic wave of Eq.(5.23), we find for the dynamics of atwo=level atom interacting with a coherent driving field, normalized to afrequency, the Rabi-Frequency

d

dtd = j2∆d+ j

Ωr

2· w (5.47)

d

dtw = +jΩ∗rd+ c.c (5.48)


5.4 Energy- and Phase-Relaxation

In reality it is extremely difficult to completely isolate an atom from itsenvironment. For example, an atom in free space does not only interact withan external electric field that is intentionally incident upon an atom butalso with the electric field from all the free space modes of the surroundingelectromagnetic field. If the atom is in a solid, it interacts with the latticevibrations of the solid. This random interaction leads to a thermalizationand decoherence of the atom. As an example for the interaction of an atomwith its environment in thermal equilibrium, we consider the interactionwith the free space electromagnetic field, that is in thermal equilibrium at agiven temperature T, because we studied already the statistics of the energyfluctuations of that field in section 3.1, i.e. black body radiation. Now,we consider, that the electric field amplitude in the Bloch equations (5.47)and (5.48) is a random quantitiy and represents the field of the black bodyradiation

Ωbbr(t) =M∗

eg · ebb~

Ebb(t), (5.49)

Where Ebb(t) is the random field and ebb the random polarization of the blackbody radiation. Note, that in the process of deriving the Rabi frequency thefield is related to the field at the transition frequency ω ≈ ωeg. This thermalRabi frequency is a random process with autocorrelation function

hΩbbr(t+ τ) ∗Ωbbr(t)ith =*¯¯M∗

eg · ebb~

¯¯2+hEbb(t+ τ)Ebb(t)i . (5.50)

The details of the computation of this autocorrelation function and the ap-proximations used can be found in Appendix A, the result is

hΩbbr(t+ τ) ∗Ωbbr(t)ith =2

T1δ(τ), (5.51)

with1

T1=

¯Meg

¯23~2

2ω3eg~πc3

µnth(ωeg) +

1

2

¶, (5.52)

and nth(ωeg) = 1/(exp(~ωeg/kT )− 1) (5.53)

This result can be easily interpreted. The first factor |Meg|23~ comes from

the average of the projection of the dipol matix element onto a unit vector

5.4. ENERGY- AND PHASE-RELAXATION 257

when averaged over every possible polarization direction. The second factor4ω3eg~πc3

¡nth(ωeg) +

12

¢comes from expressing the power spectral density of the

electric field amplitudes at the transisition frequency ω ≈ ωeg by the spectralenergy density of the black body radiation according to section 3.1. However,we did not only include in the energy the part due to the thermal photonpopulation of the mode, but also its ground state energy ~ωeg/2, which wedid not include in section 3.1 but found when treating the harmonic oscillatorquantum mechanically in section 4.4.2. Thus even at temperature T → 0, 1

T1stays finite, The white noise property helps us to find an equation of motionfor the inversion in the presence of this fluctuating field. The solution to Eq.(5.47) can be written as

d(t) =j

2

Z t

−∞e(j∆−γ)(t−t

0)Ωbbr(t0) · w(t0)dt0. (5.54)

Here, γ is a decay term that ensures convergence of integrals. As we will see,the final results will not depend on γ. The equation for the inversion is then

d

dtw = −1

2

Z t

−∞e(j∆−γ)(t−t

0)Ω∗bbr(t)Ωbbr(t0) · w(t0)dt0 + c.c (5.55)

Not, after performing this statistical average, the inversion w does no longerdescribe the inversion of a specific atom but rather the average inversion ofan ensemble of such atoms. We perform now an average over the randomthermal field and obtain

d

dtw = −1

2

Z t

−∞e(j∆−γ)(t−t

0) hΩ∗bbr(t)Ωbbr(t0)ith · w(t0)dt0 + c.c (5.56)

= −12

Z t

−∞e(j∆−γ)(t−t

0) 2

T1δ(t− t0) · w(t0)dt0 + c.c (5.57)

= − 1

2T1w(t) + c.c = − 1

T1w(t). (5.58)

Note, the evaluation of the integral over the delta-function is ambiguous,since the singularity of the delta-function was at the integration boundary.However, the delta-function is the limit of a regular function. Here, we usedthe limit of a symmetric function, which contributes 1/2 after the executionof the integral. The random field fluctuations of the electromagnetic vacuum


lead to an exponential decay of the population inversion. Note, a similarcomputation could be carried out, integrating first the equation for the in-version and substituting it into the equation for the dipole moment, whichwill lead to an exponential decay term according to

d

dtd = j∆d− 1

2T1d. (5.59)

As for the inversion, this equation describes now the average dipole momentin an ensemble of identical atoms, rather then for a specific one. Note, sofar we only averaged the equations of motion over the random excitationfield assuming that no interactions existed between the random field and theatomic system at the start of the integration. This is not true, of course insteady state the atomic system should also be in thermal equilibrium withthe thermal population inversion w0. To account for this fact, the equationfor the inversion has to be changed to

d

dtw = −w − w0

T1. (5.60)

This rate equation for the population difference can be compared with ourphenomenolocial discussion of how thermal equilibrium between thermal ra-diation and a two-level system is reached using Einsteins A and B coefficientsin section 3.4. The approach here and back then should give the same result.In section 3.4 we used as variables the densities for the ground and excitedstate n1 and n2. But normalization of these densities to the total density ofatoms gives the propability to be in the excited or ground state, respectively.Eq.(3.65), which is given here again

dn1dt

= −dn2dt=1

τ sp[(n2−n1) hsi+ n2] . (5.61)

can be transfered to

dPg

dt= −dPe

dt=1

τ sp

£(Pe−Pg)nth+Pe

¤. (5.62)

or rewritten as

dPg

dt= −dPe

dt=ΓePe − ΓaPg . (5.63)

5.4. ENERGY- AND PHASE-RELAXATION 259

with abbreviations

Γe =1

τ sp(nth + 1), (5.64)

Γa =1

τ spnth. (5.65)

see Figure 5.3.

EPe

Eg

Ee

Pg

Figure 5.3: Two-level atom with transistion rates due to induced and spon-taneous emission and absorption.

For the inversion we obtain then

d

dtw =

d

dtPe −

d

dtPg =

2

τ sp

£(Pe−Pg)nth+Pe

¤(5.66)

=−2τ sp

∙wnth+

w + 1

2

¸= −2nth + 1

τ sp

∙w+

1

2nth + 1

¸. (5.67)

Note, here we used that Pe+Pg = 1 and, therefore, Pe =w+12. Comparing

coefficients between Eqs.(5.60) and (5.67) we find

1

T1=

2nth + 1

τ sp= Γe + Γa (5.68)

w0 =Γa − ΓeΓa + Γe

=−1

2nth + 1= − tanh

µ~ωeg

2kT

¶. (5.69)


For zero temperature the decay time T1 approaches the spontaneous lifetimeof the atom due to the zero-point fluctuations of the electromagnetic field

1

T1=

1

τ sp=

¯Meg

¯2ω3eg

3π~c3.

This is an expression for the spontaneous lifetime of an atom in terms of thedipole matrix element and the density of modes in the electromagnetic fieldat the transition frequency ωeg.

In summary, the equation for the dipole moment d and the inversion wdue to its interaction with the environment can be written as

d = (j (ωeg − ω)− 1

T2)d, (5.70)

w = −w − w0T1

, (5.71)

The time constant T1 denotes the energy relaxation in the two-level systemand T2 the phase relaxation. T2 is the correlation time between amplitudesce and cg. The coherence between the excited and the ground state describedby the dipole moment is destroyed by the interaction of the two-level systemwith the environment.In this basic model, the energy relaxation rate is exactly half the phase

relaxation rate orT2 = 2T1. (5.72)

The atoms in a real medium do not only interact with the electromagneticfield, but also with phonons, i.e. acoustic vibrations of the host lattice orwith themselves through collisions in a gas laser. All these processes must beconsidered when determining the energy and phase relaxation rates. Thusthere might not only be radiative transistions that lead to a finite energyrelaxation time T1. Some of the processes are elastic, i.e. there is no energyrelaxation but only the phase is influenced during the collision. Therefore,these processes reduce T2 but have no influence on T1. In real systems thephase relaxation time is most often much shorter than twice the energy re-laxation time.

T2 ≤ 2T1. (5.73)

5.5. THE BLOCH EQUATIONS 261

If the inversion deviates from its equilibrium value, w0, it relaxes backinto equilibrium with a time constant T1. Eq. (5.69) shows that for all tem-peratures T > 0 the inversion is negative, i.e. the lower level is strongerpopulated than the upper level. Thus with incoherent thermal light, inver-sion in a two-level system cannot be achieved. Inversion can only be achievedby pumping with incoherent light, if there are more levels and subsequentrelaxation processes into the upper laser level. Due to these relaxation pro-cesses the rate Γa deviates from the equilibrium expression (5.65), and ithas to be replaced by the pump rate Λ. If the pump rate Λ exceeds Γe, theinversion corresponding to Eq. (5.69) becomes positive,

w0 =Λ− ΓeΛ+ Γe

. (5.74)

If we allow for artificial negative temperatures, we obtain with T < 0 for theratio of relaxation rates

ΓeΓa=1 + nthnth

= e~ωegkT < 1. (5.75)

Thus the pumping of the two-level system drives the system away from ther-mal equilibrium. Now, we have a correct description of an ensemble of atomsin thermal equilibrium with its environment, which is a much more realisticdescription of media especially of typical laser media.

5.5 The Bloch Equations

Thus, the total dynamics of the two-level system including the pumping anddephasing processes from Eqs.(5.70) and (5.71) is given by

d = −( 1T2− j (ωeg − ω))d+ j

Ωr

2w, (5.76)

w = −w − w0T1

+ jΩ∗r d− jΩr d∗. (5.77)

These equations are called the Bloch Equations. They describe the dynamicsof a statistical ensemble of two-level atoms interacting with a classical electricfield. Together with the Maxwell-Equations, where the polarization of themedium is related to the expectation value of the dipole moment of theatomic ensemble these result in the Maxwell-Bloch Equations.


5.6 Dielectric Susceptibility and Saturation

The Bloch Equations are nonlinear. However, for moderate field strengthE0, i.e. the magnitude of the Rabi-frequency is much smaller than the opticalfrequency, |Ωr| << ω, the inversion does not change much within an opticalcycle of the field. We assume that the inversion w of the atom will only beslowly changing and it adjusts itself to a steady state value ws. For a constantfield strength E0 Eqs.(5.76) and (5.77) reach the steady state values

ds =j

2~

³M∗

eg · e´ws

1/T2 + j(ω − ωeg)E0 (5.78)

ws =w0

1 + T1~2

1/T2 |M∗eg·e|2

(1/T2)2+(ωeg−ω)2 |E0|2. (5.79)

We introduce the normalized lineshape function, which is in this case aLorentzian,

L(ω) =(1/T2)

2

(1/T2)2 + (ωeg − ω)2, (5.80)

and connect the square of the field |E0|2 to the intensity I of a propagatingplane wave, according to Eq. (2.37), I = 1

2ZF|E0|2,

ws =w0

1 + IIsL(ω)

. (5.81)

Thus the stationary inversion depends on the intensity of the incident light.Therefore, w0 is called the unsaturated inversion, ws the saturated inversionand Is,with

Is =

∙2T1T2ZF

~2|M∗

eg · e|2¸−1

, (5.82)

is the saturation intensity. The expectation value of the dipole operator(5.22) is then given by D

dE=Megd ejωt + c.c. (5.83)

Multiplication with the number of atoms per unit volume, N, relates thedipole moment of the atom to the macroscopic polarization P . As the electricfield also the polarization can be written in terms of complex quantities

5.6. DIELECTRIC SUSCEPTIBILITY AND SATURATION 263

P (t) =1

2

³P 0e

jωt + P∗0e−jωt

´(5.84)

= NMegds ejωt + c.c. (5.85)

or

P 0 = 2NMegds, (5.86)

With the definition of the complex susceptibility

P 0 = 0χ(ω)eE0, (5.87)

and comparison with Eqs. (5.86) and Eq. (5.78), we obtain for the linearsusceptibility of the medium

χ(ω) =MegM+eg

jN

~ 0

ws

1/T2 + j(ω − ωeg), (5.88)

which is a tensor. In the following we assume that the direction of the atomis random, i.e. the alignment of the atomic dipole moment, Meg, and theelectric field is random. Therefore, we have to average over the angle enclosedbetween the electric field of the wave and the atomic dipole moment, whichresults in⎛⎝ MegxM

∗egx MegxM

∗egy MegxM

∗egz

MegyM∗egx MegyM

∗egy MegyM

∗egz

MegzM∗egx MegzM

∗egy MegzM

∗egz

⎞⎠ =

⎛⎝ M2egx 0 0

0 M2egy 0

0 0 M2egz

⎞⎠ =1

3|Meg|2 1.

(5.89)How to arrive at this average over the orientation is also discussed in Ap-pendix A. Thus, for homogeneous and isotropic media the susceptibility ten-sor shrinks to a scalar

χ(ω) =1

3|Meg|2

jN

~ 0

ws

1/T2 + j(ω − ωeg). (5.90)

Real and imaginary part of the susceptibility

χ(ω) = χ0(ω) + jχ00(ω) (5.91)


are then given by

χ0(ω) = − |Meg|2NwsT22 (ωeg − ω)

3~ 0L(ω), (5.92)

χ00(ω) =|Meg|2NwsT2

3~ 0L(ω). (5.93)

If the incident radiation is weak, i.e.

I

IsL(ω)¿ 1 (5.94)

we obtain ws ≈ w0. For optical transitions there is no thermal excitation ofthe excited state and w0 = −1. For an inverted system, w0 > 0, the real andimaginary parts of the susceptibility are shown in Fig. 5.4.

The shape of the susceptibility computed quantum mechanically com-pares well with the classical susceptibility (2.44) derived from the harmonicoscillator model close to the transistion frequency ωeg for a transition withreasonably high Q = T2ωeg. Note, the quantum mechanical susceptibility isidentical to the complex Lorentzian introduced in Eq.(2.96). There is anappreciable deviation, however, far away from resonance. Far off resonancethe rotating wave approximation should not be used.

The physical meaning of the real and imaginary part of the susceptibilityis of course identical to section 2.1.8. The propagation constant k of a TEM-wave in such a medium is related to the susceptibility by

k = ωpμ0 0(1 + χ(ω)) ≈ k0

µ1 +

1

2χ(ω)

¶, with k0 = ω

√μ0 0 (5.95)

for |χ| ¿ 1. Under this assumption we obtain

k = k0(1 +χ0

2) + jk0

χ00

2. (5.96)

5.7. RATE EQUATIONS AND CROSS SECTIONS 265

1.0

0.5

0.0

χ''(

ω) /

χ '' ma

x

2.01.51.00.50.0ω / ω eg

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

χ'( ω) / χ''max

T 2 ω eg =10

Figure 5.4: Real and imaginary part of the complex susceptibility for aninverted medium ws > 0. The positive imaginary susceptibility indicatesexponential growth of an electromagnetic wave propagating in the medium.

The real part of the susceptibility contributes to the refractive index n =1 + χ0/2. In the case of χ00 < 0, the imaginary part leads to an exponentialdamping of the wave. For χ00 > 0 amplification takes place. Amplification ofthe wave is possible for w0 > 0, i.e. an inverted medium.The phase relaxation rate 1/T2 of the dipole moment determines the width

of the absorption line or the bandwidth of the amplifier. The amplificationcan not occur forever, because the amplifier saturates when the intensityreaches the saturation intensity. This is a strong deviation from the linearsusceptibility we derived from the classical oscillator model. The reason forthis saturation is two fold. First, the light can not extract more energyfrom the atoms then there is energy stored in them, i.e. energy conservationholds. Second the induced dipole moment in a two-level atom is limited bythe maximum value of the matrix element. In contrast the induced dipolemoment in a classical oscillator grows proportionally to the applied fieldwithout limits.

5.7 Rate Equations and Cross Sections

In many cases the fastest process in the atom-field interaction dynamics isthe dephasing of the dipole moment, i. e. T2 → 0. For example, in semi-


conductors T2 < 50fs. In those cases the magnitude of the dipole momentrelaxes instantaneously into the steady state and follows the slowly varyingfield envelope E0(t) electromagntic field, which evolves on a much slower timescale. We obtain with the quasi steady state solution for the dipole moment(5.78), which may now have a slow time dependence due to the slowly varyingfield envelope E0(t), for the time dependent inversion in the atomic system

w = −w(t)− w0T1

− w(t)

T1IsL(ω)I(t), (5.97)

where I(t) = |E0(t)|2 /(2ZF ) is the intensity of the electromagntic wave in-teracting with the two-level atom. In this limit the Bloch Equations arereplaced by a simple rate equation for the population. We only take careof the counting of population differences due to spontaneous and stimulatedemissions.The interaction of an atom with light at a given transition with the stream

of photons on resonance, i.e. ω = ωeg is often discribed by the mass actionlaw. That is, the number of induced transistions from the excited to theground state, is proportional to the product of the number of atoms in theexcited state and the photon flux density Iph = I/~ωeg

w|induced = −σwIph = −w

T1IsI. (5.98)

This defines an interaction cross section σ that can be expressed in terms ofthe saturation intensity as

σ =~ωeg

T1Is(5.99)

=2ωegT2ZF

~|M∗

eg · e|2. (5.100)

In this chapter, we found the most important spectroscopic quantitiesthat characterize an atomic transition, which are the lifetime of the excitedstate or often called upper-state lifetime or longitudinal lifetime T1, the phaserelaxation time or transverse relaxation time T2 which is the inverse half-width at half maximum of the line and the interaction cross-section σ thatonly depends on the dipole matrix element and the linewidth of the transition.


Appendix: A. Derivation of Field Autocorrelation Function

For understanding the relaxation processes in a two-level system subject to athermal radiation field, we need to calculate the following expectation value

hΩbbr(t+ τ) ∗Ωbbr(t)ith =*¯¯M∗

eg · ebb~

¯¯2+hEbb(t+ τ)Ebb(t)i . (5.101)

We first compute the average ¿¯M∗

eg · ebb¯2À

,

which is the projection of a vector onto a random unit vector, see Figure 5.5.

Meg ϑ

ebb

Figure 5.5: Projection onto a unit vector and averaging over all possibleanges.

This average is over the full volume angle, whereby it does not dependon the polar angle ϕ.¿¯

M∗eg · ebb

¯2À=

1

2

Z π

0

¯M∗

eg

¯2cos2 (ϑ) sin (ϑ) dϑ (5.102)

=−16

¯M∗

eg

¯2cos3 (ϑ)

¯π0=

¯M∗

eg

¯23

·

Next is the field autocorrelation function hEbb(t+ τ)Ebb(t)i , which is theinverse Fourier transform of the power spectral density of the electric field.


For a TEM wave the electric field amplitude is directly related to the energydensity in the wave, see Table 2.1

w =1

2|E|2 . (5.103)

Thus the power spectral density of the electric field of a thermal noise sourceis directly related to the spectral energy density of a thermal radiator, whichis given by Planck’s law

SEE(ω) =2w (f) =

2 8π f 2

c3hf

exp hfkT− 1

, (5.104)

SEE(ω) =4π ~ω 3

c3hsi , with hsi = 1

exp hfkT− 1

. (5.105)

In the derivation of Planck’s law we only included for the energy of a mode,which is identical to a harmonic oscillator, the exchangable energy ~ω hsi ,where hsi is the average photon number. However, from the quantum me-chanical treatment of the harmonic oscillator in section 4.4.2, we know thatthe ground state energy, i.e. corresponding to zero photon number is notzero, but rather ~ω/2. If we want to describe the fluctuations in the electricfield properly also for zero temperature, we need to include the zero pointenergy, which describes the random field fluctuations in the ground state

SEE(ω) =4π ~ω 3

c3

µnth(ω) +

1

2

¶. (5.106)

As we have discussed before, and is shown in 5.6, the atom only interactsstrongly with fields in the spectral range close to the transistion frequency,ω ≈ ωeg. Therefore, we replace the power spectral density for the fields inthis frequency range by

SEE(ω) =4π ~ω 3eg

c3

µnth(ωeg) +

1

2

¶, (5.107)

since only those modes will interact with the atom. The constant spectraldensity simplifies thel field autorcorrelation function in the time domain to

hEbb(t+ τ)Ebb(t)i =4π ~ω 3eg

c3

µnth(ωeg) +

1

2

¶δ (τ) , (5.108)

with nth(ωeg) =1

exp ωegkT− 1


hΩbbr(t+ τ) ∗Ωbbr(t)ith =2

T1δ(τ), (5.109)

with1

T1=

¯Meg

¯23~2

2ω3eg~πc3

µnth(ωeg) +

1

2

¶, (5.110)

Bibliography

[1] I. I. Rabi: "Space Quantization in a Gyrating Magnetic Field," Phys.Rev. 51, 652-654 (1937).

[2] B. R. Mollow, "Power Spectrum of Light Scattered by Two-Level Sys-tems," Phys. Rev 188, 1969-1975 (1969).

[3] P. Meystre, M. Sargent III: Elements of Quantum Optics, Springer Verlag(1990).

[4] L. Allen and J. H. Eberly: Optical Resonance and Two-Level Atoms,Dover Verlag (1987).

271

272 BIBLIOGRAPHY

Chapter 6

Lasers

After having derived the quantum mechanically correct suszeptibility for aninverted atomic system that can provide gain, we can use the rate equationsto study the laser and its dynamics. After discussing the laser concept brieflywe will investigate various types of gain media, gas, liquid and solid-state,that can be used to construct lasers and optical amplifiers. Then the dy-namics of lasers, threshold behavior, steady state behavior and relaxationoscillations are discussed. A short introduction in the generation of high en-ergy and ultrashort laser pulses using Q-switching and mode locking will begiven at the end.

6.1 The Laser (Oscillator) Concept

Since the invention of the vacuum amplifier tube by Robert von Lieben andLee de Forest in 1905/06 it was known how to amplify electromagnetic wavesover a broad wavelength range and how to build oscillators with which elec-tromagnetic waves could be generated. This was extended into the millimeterwave region with advances in amplifier tubes and later solid-state devices suchas transistors. Until the 1950’s thermal radiation sources were mostly usedto generate electromagnetic waves in the optical frequency range. The gen-eration of coherent optical waves was only made possible by the Laser. Thefirst amplifier based on discrete energy levels (quantum amplifier) was theMASER (Microwave Amplification by Stimulated Emission of Radiation),which was invented by Gordon, Townes and Zeiger 1954. In 1958 Schawlowand Townes proposed to extend the MASER principle to the optical regime.

273

274 CHAPTER 6. LASERS

The amplification should arise from stimulated emission between discrete en-ergy levels that must be inverted, as discussed in the last section. Amplifiersand oscillators based on this principle are called LASER (Light Amplificationby Stimulated Emission of Radiation). Maiman was first to demonstrate alaser. It was based on the solid-state laser material Ruby.

Figure 6.1: Theodore Maiman with the first Ruby Laser in 1960 and a crosssectional view of the first device [4].

The first gas laser, a HeNe-Laser followed in 1961. It is a gas laser builtby Ali Javan at MIT, with a wavelength of 632.8 nm and a linewidth of only10kHz.The basic principle of an oscillator is a feedback circuit that is unstable,

i.e. there is positive feedback at certain frequencies or certain frequencyranges, see Figure 6.2. It is the feedback circuit that determines the frequencyof oscillation. Once the oscillation starts, the oscillation will build up to anintensity approaching, or even surpassing, the saturation intensity of theamplifier medium by many times, until the amplifier gain is reduced to avalue equal to the losses that the signal experiences after one roundtrip inthe feedback loop, see Figure 6.3

6.1. THE LASER (OSCILLATOR) CONCEPT 275

Figure 6.2: Principle of an oscillator circuit: an amplifier with positive feed-back [6] p. 495.

Figure 6.3: Saturation of amplification with increasing signal power leads toa stable oscillation [6], p. 496.

In the radio frequency range the feedback circuit can be an electronicfeedback circuit. At optical frequencies we use an optical resonator, whichis in most cases well modeled as a one-dimensional Fabry-Perot resonator,which we analysed in depth in section 6.6. We already found back then thatthe transfer characterisitcs of a Fabry-Perot resonator can be understood asa feedback structure. All we need to do to construct an oscillator is provideamplification in the feedback loop, i.e. to compensate in the resonator foreventual internal losses or the losses due to the output coupling via the mir-rors of the Fabry-Perot, see Figure 6.4.We have already discussed in section


2.5.3 various optical resonators, which have Gaussian beams as the funda-mental resonator modes. One can also use waveguides or fibers that havesemitransparent mirrors at its ends or form rings as laser resonators. In thelatter ones output coupling of radiation is achieved with waveguide or fibercouplers in the rings.Today, lasers generating light continuosly or in the form of long, nanosec-

ond, or very short, femtosecond pulses can be built. Typically these lasersare Q-switched or mode-locked, respectively. The average power level canvary from microwatt to kilowatts and peak powers over femtosecond pulsedurations can reach petawatts. The estimated average power consumptionof the world in 2001 was about 2 Terawatt.

Figure 6.4: A laser consists of an optical resonator where the internal lossesand/or the losses due to partially reflecting mirrors are compensated by again medium inside the resonator [6], p. 496.

6.2 Laser Gain Media

Important characteristics of laser gain media are how inversion can be achievedand what the spectroscopic parameters of the transistion involved in the laserprocess are, i.e. upperstate lifetime, τL = T1, linewdith ∆fFWHM = 2

T2and

the cross-section for stimulated emission.

6.2.1 Three and Four Level Laser Media

As we discussed before inversion can not be achieved in a two level systemby optical pumping with incoherent light. The coherent regime is typicallyinaccesible by typcial optical pump sources. Inversion by optical pumping

6.2. LASER GAIN MEDIA 277

can only be achieved when using a three or four-level system, see Figures 6.5and 6.6

0

1

2

N

N

N 2

1

0

γ

γ21

10

Rp

0

1

2

N

N

N 2

1

0

γ

γ21

10

Rp

a) b)

Figure 6.5: Three-level laser medium.

3

0

1

2

N

N

N

N

3

2

1

0

γ32

21

10

Rp γ

γ

Figure 6.6: Four-level laser medium.

If the medium is in thermal equilibrium, typically only the ground state isoccupied. By optical pumping with an intense lamp (flash lamp), or anotherlaser, one can pump a significant fraction of the atoms from the ground statewith population N0 into the excited state N2, or N1 for the three level laseroperating according to the schemes shown in Figure 6.5 (a) and (b) or N3

in the case of the four level laser, see Figure 6.6. For the cases depicted in


277 (a) and Figure 6.6, if the relaxation rate γ10 is very fast compared toγ21, inversion can be achieved, i.e. N2 > N1. For the four level laser therelaxation rate γ32 should also be fast in comparison to γ21. These systemsare easy to analyze in the rate equation approximation, where the dipolemoments are already adiabatically eliminated. For example, for the threelevel system in Figure 6.5 a). we obtain the rate equations of the three levelsystem in analogy to the two-level system

d

dtN2 = −γ21N2 − σ21 (N2 −N1) Iph +Rp (6.1)

d

dtN1 = −γ10N1 + γ21N2 + σ21 (N2 −N1) Iph (6.2)

d

dtN0 = γ10N1 −Rp (6.3)

Here, σ21 is the cross section for stimulated emission between the levels 2and 1 and Iph is the photon flux at the transition frequency f21.In mostcases, there are plenty of atoms available in the ground state such that op-tical pumping can never deplete the number of atoms in the ground stateN0 = const. That is why we can assume a constant pump rate Rp. If therelaxation rate γ10 is much faster than γ21 and the number of possible stim-ulated emission events that can occur σ21 (N2 −N1) Iph, then we can setN1 = 0 and obtain only a rate equation for the upper laser level

d

dtN2 = −γ21

µN2 −

Rp

γ21

¶− σ21N2 · Iph. (6.4)

This equation is identical to the equation for the inversion of the two-levelsystem, see Eq.(5.97). Here, Rp

γ21is the equilibrium upper state population in

the absence of photons, γ21 =1τLis the inverse upper state lifetime due to

radiative and non radiative processes.Note, a similar analysis can be done for the three level laser operating

according to the scheme shown in Figure 6.5 (b). Then the relaxation ratefrom level 3 to level 2, which is now the upper laser level has to be fast. Butin addition the optical pumping must be so strong that essentially all theground state levels are depleted. Undepleted groundstate populations wouldalways lead to absorption of laser radiation.In the following we want to discuss the electronic structure of a few often

encountered laser media. A detail description of laser media can be found in[7].

6.3. TYPES OF LASERS 279

6.3 Types of Lasers

6.3.1 Gas Lasers

Helium-Neon Laser

The HeNe-Laser is the most widely used noble gas laser. Lasing can beachieved at many wavelength 632.8nm (543.5nm, 593.9nm, 611.8nm, 1.1523μm,1.52μm, 3.3913μm). Pumping is achieved by electrical discharge, see Figure6.7.

Figure 6.7: Energy level diagram of the transistions involved in the HeNelaser [9].

The helium atoms are excited to a metastable state by electron impactthrough an electric discharge occuring at about 1000V through electrodes in


a vaccum tube. The energy is then transfered to Neon by collisions. Thefirst HeNe laser operated at the 1.1523μm line [8]. HeNe lasers are used inmany applications such as interferometry, holography, spectroscopy, barcodescanning, alignment and optical demonstrations.

Argon and Krypton Ion Lasers

Similar to the HeNe-laser the Argon ion gas laser is pumped by electric dis-charge and emitts light at wavelength: 488.0nm, 514.5nm, 351nm, 465.8nm,472.7nm, 528.7nm. It is used in applications ranging from retinal photother-apy for diabetes, lithography, and pumping of other lasers.The Krypton ion gas laser is analogous to the Argon gas laser with wave-

length: 416nm, 530.9nm, 568.2nm, 647.1nm, 676.4nm, 752.5nm, 799.3nm.Also pumped by electrical discharge. When mixed with argon, it can beused as a "white-light" laser for light shows.

Carbon Lasers

In the carbon dioxide (CO2) gas laser the laser transistions are related tovibrational-rotational excitations of the CO2-molecule. CO2 lasers are highlyefficient approaching 30%. The main emission wavelengths are 10.6μm and9.4μm. They are pumped by transverse (high power) or longitudinal (lowpower) electrical discharge. It is heavily used in the material processingindustry for cutting, and welding of steel and in the medical area for surgery.Carbon monoxide (CO) gas laser: Wavelength 2.6 - 4μm, 4.8 - 8.3μm

pumped by electrical discharge are also used in material processing such asengraving and welding and in photoacoustic spectroscopy. Output powers ashigh as 100kW have been demonstrated.

6.3.2 Excimer Lasers:

Chemical lasers emitting in the UV: 193nm (ArF), 248nm (KrF), 308nm(XeCl), 353nm (XeF) excimer (excited dimer). These are molecules thatexist only if one of the atoms is electronically excited. Without excitationthe two atoms repell each other. Thus the electronic groundstate is not stableand is therefore not populated, which is ideal for laser operation. These lasersare used for ultraviolet lithography in the semiconductor industry and lasersurgery.


6.3.3 Dye Lasers:

The laser gain medium are organic dyes in solution of ethyl, methyl alcohol,glycerol or water. These dyes can be excited optically using for exampleArgon lasers that emit at 390-435nm (stilbene), 460-515nm (coumarin 102),570-640 nm (rhodamine 6G) and many others. These lasers have been widelyused in research and spectroscopy because of there wide tuning ranges. Un-fortunately, dyes are carcinogenic and as soon as tunable solid state lasermedia became available dye laser became extinct.

6.3.4 Solid-State Lasers

Ruby Laser

The first laser was indeed a solid-state laser: Ruby emitting at 694.3nm [5].

Figure 6.8: Energy level diagram for Ruby, [2], p. 13.

Ruby consists of the naturally formed crystal of aluminum oxide (Al2O3)called corundum. In that crystal some of the Al3+ ions are replaced byCr3+ ions. Its the chromium ions that give Ruby the pinkish color, i.e. its


flourescence, which is related to the laser transisitons, see the level structurein Figure 6.8. Ruby is a three level laser. Today, for the manufacturingof ruby as a laser material, artificially grown crystals from molten materialwhich crystalizes in the form of sapphire is used. The liftetime of the upperlaser level is 3ms. Pumping is usually achieved with flashlamps, see Figure6.1.

Neodymium YAG (Nd:YAG)

Neodymium YAG consists of Yttrium-Aluminium-Garnet (YAG) Y3Al5O12in which some of the Y3+ ions are replaced by Nd3+ ions. Neodymium is arare earth element, where the active electronic states are shielded inner 4fstates. Nd:YAG is a four level laser, see Figure 6.9.

Figure 6.9: Energy level diagram for Nd:YAG, [3], p. 370.

The main emission of Nd:YAG is at 1.064μm. Another line with con-siderable less gain is at 1.32μm. Initially Nd:YAG was flashlamp pumped.Today, much more efficient pumping is possible with laser diodes and diodearrays. Diode pumped versions which can be very compact and efficient arebecoming a competition for CO2-lasers in material processing, range finding,surgery, pumping of other lasers in combination with frequency doubling toproduce a green 532nm beam).Neodymium can also be doped in a host of other crystals such as YLF

(Nd:YLF) emitting at 1047μm, YVO4(Nd:YVO4) emitting at 1.064μm, glass


(Nd:Glass) at 1.062μm (Silicate glasses), 1.054μm (Phosphate glasses). Glasslasers have been used to build extremely high power (Petawatt), high energy(Megajoules) multiple beam systems for inertial confinement fusion. The bigadvantage of glass is that it can be fabricated on meter scale which is hardor even impossible to do with crystalline materials.Other rare earth elements are Er3+, Tm3+, Ho3+, Er3+, which have em-

mission lines at 1.53μm and in the 2-3μm range.

Ytterbium YAG (Yb:YAG)

Ytterbium YAG is a quasi three level laser, see Figure 283 emitting at1.030μm. The lower laser level is only 500-600cm−1 (60meV) above theground state and is therefore at room temperature heavily thermally popu-lated. The laser is pumped at 941 or 968nm with laser diodes to provide thehigh brighness pumping needed to achieve gain.

Figure 6.10: Energy level diagram of Yb:YAG, [3], p. 374.

Yb:YAG has many advantages over other laser materials:

• Very low quantum defect, i.e. difference between the photon energy


necessary for pumping and photon energy of the emitted radiation,(hfP − hfL) /hfP ˜ 9%.

• long radiative lifetime of the upper laser level, i.e. much energy can bestored in the crystal.

• high doping levels can be used without upper state lifetime quenching

• broad emission bandwidth of ∆fFWHM = 2.5THz enabling the genera-tion of sub-picosecond pulses

• with cryogenic cooling Yb:YAG becomes a four level laser.

Due to the low quantum defect and the good thermal properties of YAG,Yb:YAG lasers approaching an optical to optical efficiency of 80% and a wallplug efficiency of 40% have been demonstrated.

Titanium Sapphire (Ti:sapphire)

In contrast to Neodymium, which is a rare earth element, Titanium is atransition metal. The Ti3+ ions replace a certain fraction of the Al3+ ions insapphire (Al2O3). In transistion metal lasers, the laser active electronic statesare outer 3s electrons which couple strongly to lattice vibrations. These lat-tice vibrations lead to strong line broadening. Therefore, Ti:sapphire has anextremely broad amplification linewidth ∆fFWHM ≈ 100THz. Ti:sapphirecan provide gain from 650-1080nm. Therefore, this material is used in to-days highly-tunable or very short pulse laser systems and amplifiers. OnceTi:sapphire was developed it rapidly replaced the dye laser systems. Figure6.11 shows the absorption and emission bands of Ti:sapphire for polarizationalong its optical axis (π−polarization).


Figure 6.11: Absorption and flourescence spectra of Ti:sapphire, [10]

6.3.5 Semiconductor Lasers

An important class of solid-state lasers are semiconductor lasers. Dependingon the semiconductor material used the emission wavelength can be furtherrefined by using bandstructure engineering, 0.4 μm (GaN) or 0.63-1.55 μm(AlGaAs, InGaAs, InGaAsP) or 3-20 μm (lead salt). The AlGaAs basedlasers in the wavelength range 670nm-780 nm are used in compact disc play-ers and therefore are the most common and cheapest lasers in the world. Insemiconductor lasers the electronic bandstructure is exploited, which arisesfrom the periodic crystal potential, see problem set 10. The energy eigen-states can be characterized by the periodic crystal quasi momentum vectork, see Figure 6.12. Since the momentum carried along by an optical pho-ton is very small compared to the momentum of the electrons in the crystallattice, transistions of an electron from the valence band to the conductionband occur essentially vertically, see Figure 6.13 (a).


Conduction band

Valence band

Band Gap

Figure 6.12: (a) Energy level diagram of the electronic states in a crystalinesolid-state material. There is usually a highest occupied band, the valenceband and a lowest unoccupied band the conduction band. Electronics statesin a crystal can usually be characterized by their quasi momentum k. b) Thevalence and conduction band are separated by a band gap.

Optical Transition Unoccupied states

“Holes”

Figure 6.13: (a) At thermal equilibrium the valence band is occupied and theconduction band is unoccupied. Optical transistions occur vertically undermomentum conservation, since the photon momentum is negligible comparedto the momentum of the electrons. (b) To obtain amplification, the mediummust be inverted, i.e. electrones must be accumulated in the conductionband and empty states in the valence band. The missing electron behave asa positively charged particles called holes.


Inversion, i.e. electrons in the conduction band and empty states in thevalence band, holes, see Figure 6.13 (b) can be achieved by creating a pn-junction diode and forward biasing, see Figure 6.14.

P - doped N - doped

Acceptors

DonorsValence band

Conduction band

Position

Ener

gy

Figure 6.14: Forward biased pn-junction laser diode. Electrons and holesare injected into the space charge region of a pn-junction and emit light byrecombination.

When forward-biased electrons and holes are injected into the space chargeregion. The carriers recombine and emit the released energy in the form ofphotons with an energy roughly equal to the band gap energy. A sketch of atypical pn-junction diode laser is shown in Figure 6.15.

Figure 6.15: Typical broad area pn-homojunction laser, [3], p. 397.

The devices can be further refined by using heterojunctions so that the


carriers are precisely confined to the region of the waveguide mode, see Figure6.16.

Figure 6.16: a) Refractive index profile. b) transverse beam profile, and c)band structure (shematic) of a double-heterostructure diode laser, [3], p. 399.

.

6.3.6 Quantum Cascade Lasers

A new form of semiconductor lasers was predicted in the 70’s by two russianphysicists, Kazarinov and Suris, that is based only on one kind of electrical


carriers. These are most often chosen to be electrons because of there highermobility. This laser is therefore a unipolar device in contrast to the conven-tional semiconductor laser that uses both electrons and holes. the transitionsare intraband transistions. A layout of a quantum cascade laser is shown inFigure 6.17.

Figure 6.17: Quantum Cascade laser layout.

Like semiconductor lasers these lasers are electrically pumped. The firstlaser of this type was realized in 1994 by Federico Capasso’s group at BellLaboratories [9], 23 years after its theoretical prediction. The reason forthis delay is the necessary high quality semiconductor growth, which is onlypossible using advanced semiconductor growth capabilities such as molecularbeam epitaxy (MBE) and more recently metal oxide chemical vapor depos-tion (MOCVD). Lasers have been demostrated from the few THz range [13]up to the 3.5μm region.


6.4 Lasers and its spectroscopic parameters

Some of the most important spectroscopic parameters of the laser transitionsof often used laser media, some of which we just discussed, are summarizedin table 6.1.

Laser MediumWave-lengthλ0(nm)

CrossSectionσ (cm2)

Upper-St.LifetimeτL (μs)

Linewidth∆fFWHM =2T2(THz)

TypRefr.indexn

Nd3+:YAG 1,064 4.1 · 10−19 1,200 0.210 H 1.82Nd3+:LSB 1,062 1.3 · 10−19 87 1.2 H 1.47 (ne)Nd3+:YLF 1,047 1.8 · 10−19 450 0.390 H 1.82 (ne)Nd3+:YVO4 1,064 2.5 · 10−19 50 0.300 H 2.19 (ne)Nd3+:glass 1,054 4 · 10−20 350 3 H/I 1.5Er3+:glass 1,55 6 · 10−21 10,000 4 H/I 1.46Ruby 694.3 2 · 10−20 1,000 0.06 H 1.76Ti3+:Al2O3 660-1180 3 · 10−19 3 100 H 1.76Cr3+:LiSAF 760-960 4.8 · 10−20 67 80 H 1.4Cr3+:LiCAF 710-840 1.3 · 10−20 170 65 H 1.4Cr3+:LiSGAF 740-930 3.3 · 10−20 88 80 H 1.4He-Ne 632.8 1 · 10−13 0.7 0.0015 I ∼1Ar+ 515 3 · 10−12 0.07 0.0035 I ∼1CO2 10,600 3 · 10−18 2,900,000 0.000060 H ∼1Rhodamin-6G 560-640 3 · 10−16 0.0033 5 H 1.33semiconductors 450-30,000 ∼ 10−14 ∼ 0.002 25 H/I 3 - 4

Table 6.1: Wavelength range, cross-section for stimulated emission, upper-state lifetime, linewidth, typ of lineshape (H=homogeneously broadened,I=inhomogeneously broadened) and index for some often used solid-statelaser materials, and in comparison with semiconductor and dye lasers.

It is surprizing, that such a wide variety of different materials can becharacterized by essentially three spectroscopy parameters that describe toa large extend the laser dynamics properly when these materials are used aslaser gain media.

6.5. HOMOGENEOUS AND INHOMOGENEOUS BROADENING 291

6.5 Homogeneous and Inhomogeneous Broad-ening

Laser media are also distinguished by the line broadening mechanisms in-volved. Very often it is the case that the linewidth observed in the absorp-tion or emission spectrum is not only due to dephasing processes that areacting on all atoms in the same, i.e. homogenous way. For example, lat-tice vibrations that lead to a line broadening of electronic transisitions oflaser ions in the crystal act in the same way on all atoms in the crystal.Such mechanisms are called homogeneous broadening. However, It can bethat in an atomic ensemble there are groups of atoms with a different cen-ter frequency of the atomic transistion. The overall ensemble therefore mayeventually show a very broad linewidth but it is not related to actual dephas-ing mechanism that acts upon each atom in the ensemble. This is partiallythe case in Nd:silicate glass lasers, see table 6.1 and the linewidth is saidto be inhomogeneously broadened. Wether a transistion is homogenouslyor inhomogeneously broadened can be tested by using a laser to saturatethe medium. In a homogenously broadened medium the loss or gain satu-rates homogenously, i.e. the whole line is reduced. In an inhomogenouslybroadened medium spectral hole burning occurs, i.e. only that sub-group ofatoms that are sufficiently in resonance with the driving field saturate andthe others not, which leads to a spectral hole in the inversion of the atomsdepending on the center frequency of the atom. Figure 6.18 shows the impactof an inhomogeneously broadened gain medium on the continous wave out-put spectrum of a laser. Inhomogenous broadening leads to lasing of manylongitudinal laser modes because of inhomogenous saturation of the gain. Inthe homogenously broadened medium the gain saturates homogenously andonly one or a few modes located in frequency at the peak of the gain canlase. An often encountered inhomogenous broadening mechanism in gasesis doppler broadening. Due to the motion of the atoms in a gas relativeto an incident electromagnetic beam, the center frequency of each atomictransistion is doppler shifted according to its velocity by

f =³1± v

c

´f0, (6.5)

where the plus sign is correct for an atom moving towards the beam and theminus sign for a atom moving with the beam. The velocity distribution of anideal gas with atoms or molecules of mass m in thermal equilibrium is given


Figure 6.18: Laser with inhomogenously broaden laser medium (Nd:silicateglass) and homogenously broadened laser medium (Nd:phosphate glass), [14]

by the Maxwell-Boltzman distribution

p(vx) =

rm

2πkTexp

µ−mv2x2kT

¶. (6.6)

This means that p(vx)dvx is equal to the probability that the atom ormolecule has a velocity in the interval [vx, vx + dv]. Here, vx is the compo-nent of the velocity that is in the direction of the beam. If the homogenouslinewidth of the atoms is small compared to the doppler broading, we obtainthe lineshape of the inhomogenously broadened gas simply by substitutingthe velocity by the induced frequency shift due to the motion

vz = cf − f0f0

(6.7)

Then the lineshape is a Gaussian

g(f) =

rmc

2πkTf0exp

"−mc2

2kT

µf − f0f0

¶2#. (6.8)

The full width at half maximum of the line is

∆fFWHM =

r8 ln 2 kT

mc2f0. (6.9)

6.6. LASER DYNAMICS (SINGLE MODE) 293

6.6 Laser Dynamics (Single Mode)

In this section, we study the single mode laser dynamics. A laser, whenpumped over threshold, typically starts to lase in a few closely spaced lon-gitudinal modes, which are incoherent with each other and the dynamics isto a large extent similar to the dynamics of a single mode that carries thepower of all lasing modes. To do so, we complement the rate equations forthe populations in the atomic medium, that can be reduced to the populationof the upper laser level Eq.(6.4), as discussed before, with a rate equation forthe photon population in the laser mode.There are two different kinds of laser cavities, linear and ring cavities, see

Figure 6.19

Figure 6.19: Possible cavity configurations. (a) Schematic of a linear cavitylaser. (b) Schematic of a ring laser.

The laser resonators can be modelled as Fabry Perot resonators as dis-


cussed in section . Typically the laser is constructed to avoid lasing of trans-verse modes and only the longitudinal modes are of interest. The resonancefrequencies of the longitudinal modes are determined by the round trip phaseto be a multiple of 2π

φ(ωm) = 2mπ. (6.10)

neighboring modes are spaced in frequency by the inverse roundtrip time

φ(ω0 +∆ω) = φ(ω0) + TR∆ω = 2mπ. (6.11)

TR is the round trip time in the resonator, which is

TR =2∗L

νg, (6.12)

where νg is the group velocity in the cavity in the frequency range considered.L is the cavity length of the linear or ring cavity, where 2∗ = 1 for the ringcavity and 2∗ = 2 for the linear cavity. In the case of no dispersion, thelongitudinal modes of the resonator are multiples of the inverse roundtriptimeand

fm = m1

TR. (6.13)

The mode spacing of the longitudinal modes is

∆f = fm − fm−1 =1

TR(6.14)

If we assume frequency independent cavity loss and Lorentzian shaped gain(see Fig. 6.20). Initially when the laser gain is larger then the cavity loss,many modes will start to lase. To assure single frequency operation a filter(etalon) can be inserted into the laser resonator, see Figure 6.21. If the laseris homogenously broadened the laser gain will satured to the loss level andonly the mode at the maximum of the gain will lase. If the gain is nothomogenously broadend and in the absence of a filter many modes will lase.For the following we assume a homogenously broadend laser medium and

only one cavity mode is able to lase. We want to derive the equations ofmotion for th population inversion, or population in the upper laser level andthe photon number in that mode, see Figure 6.22.


Figure 6.20: Laser gain and cavity loss spectra, longitudinal mode location,and laser output for multimode laser operation.

Figure 6.21: Gain and loss spectra, longitudinal mode locations, and laseroutput for single mode laser operation.


Figure 6.22: Rate equations for a laser with two-level atoms and a resonator.

The intensity I in a mode propagating at group velocity vg with a modevolume V is related to the number of photons NL or the number densitynL = NL/V stored in the mode with volume V = AeffL by

I = hfLNL

2∗Vvg =

1

2∗hfLnLvg, (6.15)

where hfL is the photon energy and as before, 2∗ = 2 for a linear laserresonator (then only half of the photons are [propagating in one direction),and 2∗ = 1 for a ring laser (all photons propagate into the same direction). Inthis first treatment we consider the case of space-independent rate equations,i.e. we assume that the laser is oscillating on a single mode and pumpingand mode energy densities are uniform within the laser material. With theinteraction cross section σ for stimulated emission defined as

σ =hfLIsτL

, (6.16)

and Eq. (6.4) with the number of atoms in the upper laser level N2 in themode, we obtain

d

dtN2 = −

N2

τL− σvg

VN2NL +Rp. (6.17)

Here, vgnL is the photon flux, σ is the stimulated emission cross section,τL = γ21 the upper state lifetime and Rp is the pumping rate into the upperlaser level. A similar rate equation can be derived for the total number ofphotons in the mode

d

dtNL = −

NL

τ p+

σvgV

N2 (NL + 1) . (6.18)


Here, τ p is the photon lifetime in the cavity or cavity decay time. The 1 inthe term (NL + 1) in Eq.(6.18) accounts for the spontaneously photons intothe laser mode, which is equivalent to stimulated emission by one photonoccupying the mode, as we have seen in section 3.4 Eq.(3.64). Note, this isnot the spontaneous emmission into all the free space modes, which is includein the upper state lifetime τL.For a laser cavity with a semi-transparent mirror with power transmis-

sion T , see section 2.4.8, producing a power loss 2l = T per round-trip inthe cavity, the cavity decay time is 1/τ p = 2l/TR , if TR = 2∗L/vg is theroundtrip-time in linear cavity with optical length 2L or a ring cavity withoptical length L. Eventual internal losses can be treated in a similar way andcontribute to the cavity decay time. Note, the decay rate for the inversionin the absence of a field, 1/τL, is not only due to spontaneous emission intofree space modes, but is also a result of non radiative decay processes.So the two rate equations are

d

dtN2 = −N2

τL− σvg

VN2NL +Rp (6.19)

d

dtNL = −NL

τ p+

σvgV

N2 (NL + 1) . (6.20)

Experimentally, the photon number and the inversion in a laser resonator arenot the quantities directly measured. We therefore introduce the circulatingintracavity power and the roundtrip gain

P = I ·Aeff = hfLNL

TR, (6.21)

g =σvg2V

N2TR. (6.22)

The intracavity power is directly proportional to the output power from thelaser

Pout = T · P. (6.23)

From Eqs.(6.19) and (6.20) for inversion and photon number, we obtain

d

dtg = −g − g0

τL− gP

Esat(6.24)

d

dtP = − 1

τ pP +

2g

TR(P + Pvac) , (6.25)


where we introduced the following laser parameters, defined as the saturationenergy of the gain medium, Esat, the saturation power of the gain medium,Psat the "vacuum"-power, Pvac, and the small signal gain, g0, of the laser

Esat =hfLV

σvgTR=1

2∗IsAeffτL (6.26)

Psat = Esat/τL (6.27)

Pvac = hfL/TR (6.28)

g0 = 2∗Rp

2AeffστL, (6.29)

Note, the factor of two in front of gain and loss is due to the fact, thatwe defined g and l as gain and loss with respect to amplitude. Eq.(6.29)elucidates that the figure of merit that characterizes the small signal gainachievable in a laser connected with the spectroscopic parameters of thelaser gain medium is the σ · τL-product.

6.7 Continuous Wave Operation

In many technical applications, where the dimensions of the laser is largecompared to the wavelength and Pvac ¿ P ¿ Psat = Esat/τL, we can neglectthe spontaneous emission, i.e. Pvac. If the laser power is initially small, on theorder of Pvac, and the gain is unsaturated, g = g0, we obtain from Eq.(6.25),

dP

P= 2 (g0 − l)

dt

TR(6.30)

or

P (t) = Pvac e2(g0−l) t

TR . (6.31)

The laser power builts up from vaccum fluctuations, once the small signalgain surpasses the laser losses, g0 > gth = l, see Figure 6.23, until it reachesthe saturation power. Saturation sets in within the built-up time

TB =TR

2 (g0 − l)ln

Psat

Pvac=

TR2 (g0 − l)

lnAeffTRστL

. (6.32)

6.7. CONTINUOUS WAVE OPERATION 299

Figure 6.23: Built-up of laser power from spontaneous emission noise.

Some time after the built-up phase the laser reaches steady state, witha certain saturated gain and steady state power resulting from Eqs.(6.24-6.25), neglecting in the following the spontaneous emission, Pvac = 0, and forddt= 0 :

gs =g0

1 + PsPsat

= l (6.33)

Ps = Psat

³g0l− 1´, (6.34)

Figure 6.24 shows output power and gain as a function of small signal gaing0, which is proportional to the pump rate Rp. Below threshold, the outputpower is zero and the gain increases linearly with in crease pumping. Afterreaching threshold the gain stays clamped at the threshold value determinedby gain equal loss and the output power increases linearly. The thresholdcondition is again

gth = l, (6.35)

Rp,th =2lAeff

2∗στL. (6.36)

Thus the pump rate to reach threshold is proportional to the optical loss ofthe mode per roundtrip, the mode cross section (in the gain medium) andinverse proportional to the σ · τL−product.


Out

put P

ower

, Gai

n

Small signal gain g0g = =l0 gth

g= =lgth

P=0

Figure 6.24: Output power and gain of a laser as a function of pump power.

6.8 Stability and Relaxation Oscillations

How does the laser reach steady state, once a perturbation has occured?

g = gs +∆g (6.37)

P = Ps +∆P (6.38)

Substitution into Eqs.(6.24-6.25) and linearization leads to

d∆P

dt= +2

Ps

TR∆g (6.39)

d∆g

dt= − gs

Esat∆P − 1

τ stim∆g (6.40)

where 1τstim

= 1τL

¡1 + Ps

Psat

¢is the inverse stimulated lifetime. The stimulated

lifetime is the lifetime of the upper laser state in the presence of the opticalfield. The perturbations decay or grow likeµ

∆P∆g

¶=

µ∆P0∆g0

¶est. (6.41)

which leads to the system of equations (using gs = l)

A

µ∆P0∆g0

¶=

Ã−s 2 Ps

TR

− TREsat2τp

− 1τstim

− s

!µ∆P0∆g0

¶= 0. (6.42)

6.8. STABILITY AND RELAXATION OSCILLATIONS 301

There is only a solution, if the determinante of the coefficient matrix vanishes,i.e.

s

µ1

τ stim+ s

¶+

Ps

Esatτ p= 0, (6.43)

which determines the relaxation rates or eigen frequencies of the linearizedsystem

s1/2 = −1

2τ stim±

sµ1

2τ stim

¶2− Ps

Esatτ p. (6.44)

Introducing the pump parameter r = 1 + PsPsat

, which tells us how often wepump the laser over threshold, the eigen frequencies can be rewritten as

s1/2 = − 1

2τ stim

Ã1± j

s4 (r − 1)

r

τ stimτp− 1!, (6.45)

= − r

2τL± j

s(r − 1)τLτ p

−µ

r

2τL

¶2(6.46)

There are several conclusions to draw:

• (i): The stationary state (0, g0) for g0 < l and (Ps, gs) for g0 > l arealways stable, i.e. Resi < 0.

• (ii): For lasers pumped above threshold, r > 1, and long upper statelifetimes, i.e. r

4τL< 1

τp,

the relaxation rate becomes complex, i.e. there are relaxation oscilla-tions

s1/2 = −1

2τ stim± jωR. (6.47)

with a frequency ωR approximately equal to the geometric mean ofinverse stimulated lifetime and photon life time

ωR ≈s

1

τ stimτ p. (6.48)

• If the laser can be pumped strong enough, i.e. r can be made largeenough so that the stimulated lifetime becomes as short as the cavitydecay time, relaxation oscillations vanish.


The physical reason for relaxation oscillations and instabilities related toit is, that the gain reacts to slow on the light field, i.e. the stimulated lifetimeis long in comparison with the cavity decay time.

Example: diode-pumped Nd:YAG-Laser

λ0 = 1064 nm, σ = 4 · 10−19cm2, Aeff = π (100μm× 150μm) , r = 50τL = 1.2 ms, l = 1%, TR = 10ns

From Eq.(6.16) we obtain:

Isat =hfLστL

= 0.4kW

cm2, Psat = IsatAeff = 0.18 W, Ps = 9.2W

τ stim =τLr= 24μs, τ p = 1μs, ωR =

s1

τ stimτ p= 2 · 105s−1.

Figure 6.25 shows the typically observed fluctuations of the output of a solid-

Figure 6.25: Relaxation oscillations in the time and frequency domain.

state laser in the time and frequency domain. Note, that this laser has a longupperstate lifetime of several 100 μs

6.9. LASER EFFICIENCY 303

One can also define a quality factor for the relaxation oscillations by theratio of the imaginary to the real part of the complex eigen frequencies 6.46

Q =

s4τLτ p

(r − 1)r2

. (6.49)

The quality factor can be as large a several thousand for solid-state laserswith long upper-state lifetimes in the millisecond range.

6.9 Laser Efficiency

An important measure for a laser is the efficiency with which pump poweris converted into laser output power. To determine the efficiency we mustreview the important parameters of a laser and the limitations these param-eters impose.From Eq.(6.34) we found that the steady state intracavity power Ps of a

laser is

Ps = Psat

µ2g02l− 1¶, (6.50)

where 2g0 is the small signal round-trip power gain, Psat the gain saturationpower and 2l is the power loss per round-trip. Both parameters are expressedin Eqs.(6.26)-(6.29) in terms of the fundamental pump parameter Rp, στL-product and mode cross section Aeff of the gain medium. For this derivationit was asummed that all pumped atoms are in the laser mode with constantintensity over the beam cross section

2g0 = 2∗Rp

AeffστL, (6.51)

Psat =hfL2∗στL

Aeff (6.52)

The power losses of lasers are due to the internal losses 2lint and the trans-mission T through the output coupling mirror. The internal losses can be asignificant fraction of the total losses. The output power of the laser is then

Pout = T · Psat

µ2g0

2lint + T− 1¶

(6.53)


The pump power of the laser is given by

Pp = RphfP , (6.54)

where hfP is the energy of the pump photons. In discussing the efficiency ofa laser, we consider the overall efficiency

η =Pout

Pp(6.55)

which approaches the differential efficiency ηD if the laser is pumped manytimes over threshold, i.e. r = 2g0/2l→∞

ηD =∂Pout

∂Pp= η(r→∞) (6.56)

=T

2lint + TPsat

2∗

AeffhfPστL (6.57)

=T

2lint + T· hfLhfP

. (6.58)

Thus the efficiency of a laser is fundamentally limited by the ratio of outputcoupling to total losses and the quantum defect in pumping. Therefore,one would expect that the optimum output coupling is achieved with thelargest output coupler, however, this is not true as we considered the case ofoperating many times above threshold, see problem set 9.

6.10 "Thresholdless" Lasing

So far we neglected the spontaneous emission into the laser mode. This isjustified for large lasers where the density of radiation modes in the lasermedium is essentially the free space mode density and effects very close tothreshold are not of interest. For lasers with small mode volume, or for a laseroperating very close to threshold, the spontaneous emission into the lasermode can no longer be neglected and we should use the full rate equations

6.10. "THRESHOLDLESS" LASING 305

(6.24) and (6.25)

d

dtg = −g − g0

τL− gP

Esat(6.59)

d

dtP = − 1

τ pP +

2g

TR(P + Pvac) , (6.60)

where Pvac is the power of a single photon in the mode. The steady stateconditions are

gs =g0

(1 + Ps/Psat), (6.61)

0 = (2gs − 2l)P + 2gsPvac. (6.62)

Substitution of the saturated gain condition (6.61) into (6.62) and using thepump parameter r = 2g0/2l, leads to a quadratic equation for the normalizedintracavity steady state power p=Ps/Psat in terms of normalized vacuumpower pv = Pvac/Psat = στLvg/V. This equation has the solutions

p =r − 1 + rpv

2±

sµr − 1 + rpv

2

¶2+ (rpv)

2. (6.63)

where only the solution with the plus sign is of physical significance. Note,the typical value for the στL-product of the laser materials in table 6.1 isστL = 10

−23cm2s. If the volume is measured in units of wavelength cubed,V = βλ3, we obtain pv = 0.3/β for λ = 1μm, and vg = c. Figure 6.26 showsthe behavior of the intracavity power as a function of the pump parameter forvarious values of the normalized vacuum power on a linear and logarithmicscale.


10 -10

10 -8

10 -6

10 -4

10 -2 10 0

Intrac

avity

powe

r, p

2.01.51.00.5Pump parameter r

p v =10 -310 -2

10 -11

1.0

0.5

0.0

Intrac

avity

powe

r, p

2.01.51.00.50.0Pump parameter r

p v =10 -3

10 -210 -11

Figure 6.26: Intracavity power as a function of pump parameter r on a linearscale (a) and a logarithmic scale (b) for various values of the normalizedvacuum power pv.

Figure 6.26 shows that for lasers with small mode volumes, i.e. modevolumes of the size of the wavelength cubed, the threshold is no longer welldefined.

6.11 Short pulse generation by Q-Switching

The energy stored in the laser medium can be released suddenly (exponen-tially fast) when switching the laser from a high loss situation above thresholdby reducing the loss suddently, i.e. increasing the Q-value of the cavity. This

6.11. SHORT PULSE GENERATION BY Q-SWITCHING 307

can be done actively, for example by quickly moving one of the resonatormirrors in place or passively by placing a saturable absorber in the resonator[1, 8]. Hellwarth first suggest this method only one year after the invention ofthe laser. As a rough orientation for a solid-state laser the following relationfor the relevant time scales that govern the laser dynamics is generally valid

τL À TR À τ p. (6.64)

6.11.1 Active Q-Switching

(a)Losses

t

High losses, laser is below threshold

(b)Losses

t

Build-up of inversion by pumping

(c)Losses

t

In active Q-switching, the losses are reduafter the laser medium is pumped for as loupper state lifetime. Then the loss is reduand laser oscillation starts.

(d)

Gain

Losses

t

Length of pump pulse

"Q-switched" Laserpuls

Laser emission stops after the energy stothe gain medium is extracted.

Gain

Gain

Figure 6.27: Gain and loss dynamics of an actively Q-switched laser.

Fig. 6.27 shows the principle dynamics of an actively Q-switched laser. Thelaser is pumped by a pump pulse with a length on the order of the upper-state


lifetime, while the intracavity losses are kept high enough so that the laser cannot reach threshold. At this point, the laser medium acts as energy storagewith the energy slowly relaxing by spontaneous and nonradiative transitions.Intracavity loss is suddenly reduced, for example by a rotating cavity mirror.In the high-Q situation, the laser is pumped way above threshold and thelight field builts up exponentially until the pulse reaches the saturation energyof the gain medium. The gain saturates and its energy is extracted, causingthe laser to be shut off by the pulse itself.A typical actively Q-switched pulse is asymmetric: The rise time is pro-

portional to the net gain after the Q-value of the cavity is actively switchedto a high value. The light intensity grows proportional to 2g0/TR. When thegain is depleted, the fall time mostly depends on the cavity decay time τ p,see Figure 6.28. For short Q-switched pulses a short cavity length, high gainand a large change in the cavity Q is necessary. If the Q-switch is not fast,the pulse width may be limited by the speed of the switch. Typical timescales for electro-optical and acousto-optical switches are 10 ns and 50 ns,respectively

0 20161284Time (ns)

Inte

nsity

~1/τ p

~ g /TR0

Figure 6.28: Asymmetric actively Q-switched pulse.

For example, with a diode-pumped Nd:YAG microchip laser [15] using anelectro-optical switch based on LiTaO3 Q-switched pulses as short as 270 psat repetition rates of 5 kHz, peak powers of 25 kW at an average power of34 mW, and pulse energy of 6.8 μJ have been generated (Figure 6.29).

6.11. SHORT PULSE GENERATION BY Q-SWITCHING 309

Figure 6.29: Q-switched microchip laser using an electro-optic switch. Thepulse is measured with a sampling scope [15]

6.11.2 Passive Q-Switching

In the case of passive Q-switching, the intracavity loss modulation is per-formed by a saturable absorber, which introduces large losses for low inten-sities of light and small losses for high intensities.Relaxation oscillations are due to a periodic exchange of energy stored in

the laser medium by the inversion and the light field. Without the saturableabsorber these oscillations are damped. If for some reason there is too muchgain in the system, the light field can build up quickly. Especially for a low


gain cross section the back action of the growing laser field on the inversion isweak and it can grow further. This growth is favored in the presence of lossthat saturates with the intensity of the light. The laser becomes unstableand the field intensity growth as long as the gain does not saturate belowthe net loss, see Fig.6.30.

Loss

Pulse

Gain

Figure 6.30: Gain and loss dynamics of a passively Q-switched laser

The saturable absorber leads to a destabilization of the relaxation oscil-lations resulting in "giant" laser pulses. Once, the gain is depleted belowthreshold the laser shuts itself off. If continuously pumped, the gain grad-ually increases again until reaching threshold and the giant-pulse emissionstarts all over again. This system describing the photon and upper stateatomic population in the presence of a saturable absorber is similar to theLotka-Volterra equation know in population dynamics describing the cyclicbehavior of predator and prey populations.

6.12 Short pulse generation by mode locking

Q-switching is a single mode pheonomenon, i.e. the pulse build-up and decayoccurs over many round-trips via build-up of energy and decay in a single (ora few) longitudinal mode. If one could get several longitudinal modes lasingin a phase coherent fashion with resepct to each other, these modes would

6.12. SHORT PULSE GENERATION BY MODE LOCKING 311

be the Fourier components of a periodic pulse train emitted from the laser.Then a single pulse is traveling inside the laser cavity. This is called modelocking and the pulses generated are much shorter than the cavity round-triptime due to interference of the fields from many modes. The electric fieldcan be written as a superpostion of the longitudinal modes. We assume agiven polarization. The electric field in this polarization is then

E(z, t) = <"X

m

Emej(ωmt−kmz+φm)

#, (6.65a)

ωm = ω0 +m∆ω = ω0 +mπc

, (6.65b)

km =ωm

c. (6.65c)

Equation (6.65a) can be rewritten as

E(z, t) = <(ejω0(t−z/c)

Xm

Emej(m∆ω(t−z/c)+φm)

)(6.66a)

= <£A(t− z/c)ejω0(t−z/c)

¤(6.66b)

with the complex envelope

A³t− z

c

´=Xm

Emej(m∆ω(t−z/c)+φm) = complex envelope (slowly varying).

(6.67)ejω0(t−z/c) is the carrier wave (fast oscillation). Here, both the carrier and theenvelope travel with the same speed (no dispersion assumed). The envelopefunction is periodic with period

T =2π

∆ω=2

c=

L

c, (6.68)

where L is the optical round-trip length in the cavity. If we assume that Nmodes with equal amplitudes Em = E0 and equal phases φm = 0 are lasing,the envelope is given by

A(z, t) = E0

(N−1)/2Xm=−(N−1)/2

ej(m∆ω(t−z/c)). (6.69)


Withq−1Xm=0

am =1− aq

1− a, (6.70)

we obtain

A(z, t) = E0sin£N∆ω2

¡t− z

c

¢¤sin£∆ω2

¡t− z

c

¢¤ . (6.71)

The laser intensity I is proportional to E(z, t)2 averaged over one opticalcycle: I ∼ |A(z, t)|2. At z = 0, we obtain

I(t) ∼ |E0|2sin2

¡N∆ωt2

¢sin2

¡∆ωt2

¢ . (6.72)

The periodic pulses given by Eq. (6.72) have

• the period: T = 1/∆f = L/c

• pulse duration: ∆t = 2πN∆ω

= 1N∆f

• peak intensity ∼ N2|E0|2

• average intensity ∼ N |E0|2 ⇒ peak intensity is enhanced by afactor N .

If the phases of the modes are not locked, i.e. φm is a random sequencethen the intensity fluctuates randomly about its average value (∼ N |E0|2),which is the same as in the mode-locked case. Figure 6.31 shows the intensityof a modelocked laser versus time,if the relative phases of the modes to eachother is (a) constant and (b) random


Figure 6.31: Laser intensity versus time from a mode-locked laser with (a)perfectly locked phases and (b) random phases.

We are distinguishing between active modelocking, where an externalloss modulator is inserted in the cavity to generate modelocked pulses, andpassive modelocking, where the pulse is modulating the intracavity itself viaa saturable absorber.

6.12.1 Active Mode Locking

Active mode locking was first investigated in 1970 by Kuizenga and Siegman[16] and later by Haus [17] . In the approach by Haus, modelocking is treatedas a pulse propagation problem.


Figure 6.32: Actively modelocked laser with an amplitude modulator(Acousto-Optic-Modulator).

The pulse is shaped in the resonator by the finite bandwidth of the gainand the loss modulator, which periodically varies the intracavity loss ac-cording to q(t) = M (1− cos(ωMt)). The modulation frequency has to beprecisely tuned to the resonator round-trip time, ωM = 2π/TR, see Fig.6.32.The mode locking process is then described by the master equation for theslowly varying pulse envelope

TR∂A

∂T=

∙g(T ) +Dg

∂2

∂t2− l −M (1− cos(ωMt))

¸A. (6.73)

This equation can be interpreted as the total pulse shaping due to gain, lossand modulator within one roundtrip, see Fig.6.33.If we fix the gain in Eq. (6.73) at its stationary value, what ever it might

be, Eq.(6.73) is a linear p.d.e, which can be solved by separation of variables.The pulses, we expect, will have a width much shorter than the round-triptime TR. They will be located in the minimum of the loss modulation wherethe cosine-function can be approximated by a parabola and we obtain

TR∂A

∂T=

∙g − l +Dg

∂2

∂t2−Mst

2

¸A. (6.74)

Ms is the modulation strength, and corresponds to the curvature of the lossmodulation in the time domain at the minimum loss point

Dg =g

Ω2g, (6.75)

Ms =Mω2M2

. (6.76)


Figure 6.33: Schematic representation of the master equation for an activelymode-locked laser.

The differential operator on the right side of (6.74) corresponds to the Schrödinger-Operator of the harmonic oscillator problem. Therefore, the eigen functionsof this operator are the Hermite-Gaussians

An(T, t) = An(t)eλnT/TR , (6.77)

An(t) =

sWn

2n√πn!τa

Hn(t/τa)e− t2

2τ2a , (6.78)

where τa determines the pulse width of the Gaussian pulse. The width isgiven by the fourth root of the ratio between gain dispersion and modulatorstrength

τa =4

qDg/Ms. (6.79)

Note, from Eq. (6.77) we see that the gain per round-trip of each eigenmodeis given by λn (or in general the real part of λn), which is given by

λn = gn − l − 2Msτ2a(n+

1

2). (6.80)

The corresponding saturated gain for each eigen solution is given by

gn =1

1 + Wn

PsatTR

, (6.81)


whereWn is the energy of the corresponding solution and Psat = Esat/τL thesaturation power of the gain. Eq. (6.80) shows that for a given g the eigensolution with n = 0, the ground mode, has the largest gain per roundtrip.Thus, if there is initially a field distribution which is a superpostion of alleigen solutions, the ground mode will grow fastest and will saturate the gainto a value

gs = l +Msτ2a. (6.82)

such that λ0 = 0 and consequently all other modes will decay since λn < 0 forn ≥ 1. This also proves the stability of the ground mode solution [17]. Thusactive modelocking without detuning between resonator round-trip time andmodulator period leads to Gaussian steady state pulses with a FWHM pulsewidth

∆tFWHM = 2 ln 2τa = 1.66τa. (6.83)

The spectrum of the Gaussian pulse is given by

A0(ω) =

Z ∞

−∞A0(t)e

iωtdt (6.84)

=

q√πWnτae

− (ωτa)2

2 , (6.85)

and its FWHM is

∆fFWHM =1.66

2πτa. (6.86)

Therfore, the time-bandwidth product of the Gaussian is

∆tFWHM ·∆fFWHM = 0.44. (6.87)

The stationary pulse shape of the modelocked laser is due to the parabolicloss modulation (pulse shortening) in the time domain and the parabolicfiltering (pulse stretching) due to the gain in the frequency domain, see Figs.6.34 and 6.35. The stationary pulse is achieved when both effects balance.Since external modulation is limited to electronic speed and the pulse widthdoes only scale with the inverse square root of the gain bandwidth activelymodelocking typically only results in pulse width in the range of 10-100ps.


Figure 6.34: Loss modulation leads to pulse shortening in each roundtrip

Figure 6.35: The finite gain bandwidth broadens the pulse in each roundtrip.For a certain pulse width there is balance between the two processes.

For example: Nd:YAG; 2l = 2g = 10%, Ωg = π∆fFWHM = 0.65 THz,M = 0.2, fm = 100 MHz,Dg = 0.24 ps2,Ms = 4 · 1016s−1, τ p ≈ 99 ps.With the pulse width (6.79), Eq.(6.82) can be rewritten in several ways

gs = l +Msτ2a = l +

Dg

τ 2a= l +

1

2Msτ

2a +

1

2

Dg

τ 2a, (6.88)


which means that in steady state the saturated gain is lifted above the losslevel l, so that many modes in the laser are maintained above threshold.There is additional gain necessary to overcome the loss of the modulator dueto the finite temporal width of the pulse and the gain filter due to the finitebandwidth of the pulse. Usually

gs − l

l=

Msτ2a

l¿ 1, (6.89)

since the pulses are much shorter than the round-trip time. The stationarypulse energy can therefore be computed from

gs =1

1 + Ws

PLTR

= l. (6.90)

The name modelocking originates from studying this pulse formation processin the frequency domain. Note, the term

−M [1− cos(ωMt)]A

generates sidebands on each cavity mode present according to

−M [1− cos(ωMt)] exp(jωn0t)

= −M∙exp(jωn0t)−

1

2exp(j(ωn0t− ωMt))− 1

2exp(j(ωn0t+ ωMt))

¸= M

∙− exp(jωn0t) +

1

2exp(jωn0−1t) +

1

2exp(jωn0+1t)

¸if the modulation frequency is the same as the cavity round-trip frequency.The sidebands generated from each running mode is injected into the neigh-boring modes which leads to synchronisation and locking of neighboringmodes, i.e. mode-locking, see Fig.6.36

6.12.2 Passive Mode Locking

Electronic loss modulation is limited to electronic speeds. Therefore, thecurvature of the loss modulation that determines the pulse length is limited.It is desirable that the pulse itself modulates the loss. The shorter the pulsethe sharper the loss modulation and eventually much shorter pulses can bereached.


f

1-M

n0-1

MM

n0+1fn0f f

Figure 6.36: Modelocking in the frequency domain: The modulator transversenergy from each mode to its neighboring mode, thereby redistributing en-ergy from the center to the wings of the spectrum. This process seeds andinjection locks neighboring modes.

The dynamics of a laser modelocked with a fast saturable absorber canalso be easily understood using the master equation (6.73) and replacing theloss due to the modulator with the loss from a saturable absorber. For afast saturable absorber, the losses q react instantaneously on the intensity orpower P (t) = |A(t)|2 of the field

q(A) =q0

1 + |A|2PA

, (6.91)

where PA is the saturation power of the absorber. Such absorbers can bemade out of semiconductor materials or nonlinear optical effects can be usedto create artificial saturable absorption, such as in Kerr lens mode locking.There is no analytic solution of the master equation (6.73) when the

loss modulation is replaced by the absorber response (6.91). We can how-ever make expansions on the absorber response to get analytic insight. Ifthe absorber is not saturated, we can expand the response (6.91) for smallintensities

q(A) = q0 − γ|A|2, (6.92)

with the saturable absorber modulation coefficient γ = q0/PA. The constantnonsaturated loss q0 can be absorbed in the losses l0 = l + q0. The resultingmaster equation is

TR∂A(T, t)

∂T=

∙g − l0 +Dg

∂2

∂t2+ γ|A|2

¸A(T, t). (6.93)


Up to the imaginary unit, this equation is similar to a type of nonlinearSchroedinger Equation, i.e. the potential depends on the wave function itself.One finds as a possible stationary solution

As(T, t) = As(t) = A0sechµt

τ

¶. (6.94)

Note, there is

d

dxsechx = − tanhx sechx, (6.95)

d2

dx2sechx = tanh2 x sechx − sech3x,

=¡sechx− 2 sech3x

¢. (6.96)

Substitution of the solution (6.94) into the master equation (6.93), and as-suming steady state, results in

0 =

∙(g − l0) +

Dg

τ 2

∙1− 2sech2

µt

τ

¶¸+γ|A0|2sech2

µt

τ

¶¸·A0sech

µt

τ

¶. (6.97)

Comparison of the coefficients with the sech- and sech3-expressions leadsto a condition for the peak pulse intensity and pulse width, τ , and for thesaturated gain

Dg

τ 2=

1

2γ|A0|2, (6.98)

g = l0 −Df

τ 2. (6.99)

From Eq.(6.98) and with the pulse energy of a sech pulse

W =

Z +∞

−∞2|As(t)|2dt = 2|A0|2τ , (6.100)

follows

τ =4Dg

γW. (6.101)


gs(W ) =g0

1 + WPLTR

(6.102)

Equation (6.99) together with (6.101) determines the pulse energy

gs(W ) =g0

1 + WPLTR

= l0 −Dg

τ 2

= l0 −(γW )2

16Dg(6.103)

Figure 6.37 shows the time dependent variation of gain and loss in a lasermodelocked with a fast saturable absorber on a normalized time scale.

Figure 6.37: Gain and loss in a passively modelocked laser using a fast sat-urable absorber on a normalized time scale x = t/τ . The absorber is assumed

to saturate linearly with intensity according to q(A) = q0³1− |A|2

A20

´.

Here, we assumed that the absorber saturates linearly with intensity upto a maximum value q0 = γA20. If this maximum saturable absorption iscompletely exploited, see Figure 6.38.


Figure 6.38: Saturation characteristic of an ideal saturable absorber

The minimum pulse width achievable with a given saturable absorptionq0 results from Eq.(6.98)

Dg

τ 2=

q02, (6.104)

to be

τ =

r2gsq0

1

Ωg. (6.105)

Note that in contrast to active modelocking the achievable pulse width is nowscaling with the inverse gain bandwidth. This gives much shorter pulses.Figure 6.37 can be interpreted as follows: In steady state, the saturatedgain is below loss, by about one half of the exploited saturable loss beforeand after the pulse. This means that there is net loss outside the pulse,which keeps the pulse stable against growth of instabilities at the leadingand trailing edge of the pulse. If there is stable mode-locked operation, theremust always be net loss far away from the pulse, otherwise, a continuouswave signal running at the peak of the gain would experience more gain thanthe pulse and would break through. From Eq.(6.98) it follows, that one thirdof the exploited saturable loss is used up during saturation of the aborberand actually only one sixth is used to overcome the filter losses due to thefinite gain bandwidth. Note, there is a limit to the mimium pulse width.This limit is due to the saturated gain (6.99), gs = l + 1

2q0. Therefore, from

Eq.(6.105), if we assume that the finite bandwidth of the laser is set by thegain, we obtain for q0 À l

τmin =1

Ωg(6.106)


for the linearly saturating absorber model. This corresponds to mode lockingover the full bandwidth of the gain medium, as for a sech-shaped pulse, thetime-bandwidth product is 0.315, and therefore,

∆fFWHM =0.315

1.76 · τmin=

Ωg

1.76 · π . (6.107)

As an example, for the Ti:sapphire laser this corresponds to Ωg = 240 THz,τmin = 3.7 fs, τFWHM = 6.5 fs, which is in good agreement with experimen-tally observed results [19].This concludes the introdcution into ultrashort pulse generation by mode

locking.

Bibliography

[1] R. W. Hellwarth, Eds., Advances in Quantum Electronics, ColumbiaPress, New York (1961).

[2] A. E. Siegman, ”Lasers,” University Science Books, Mill Valley, Califor-nia (1986).

[3] O. Svelto, "Principles of Lasers," Plenum Press, NY 1998.

[4] http://www.llnl.gov/nif/library/aboutlasers/how.html

[5] T. H. Maimann, "Stimulated optical radiation in ruby", Nature 187,493-494, (1960).

[6] B.E.A. Saleh and M.C. Teich, "Fundamentals of Photonics," JohnWileyand Sons, Inc., 1991.

[7] M. J. Weber, "Handbook of Lasers", CRC Press, 2000.

[8] A. Javan, W. R. Bennett and D. H. Herriott, "Population Inversion andContinuous Optical Maser Oscillation in a Gas Discharge Containing aHe-Ne Mixture," Phys. Rev. Lett. 6, (1961).

[9] W. R. Bennett, Applied Optics, Supplement 1, Optical Masers, 24,(1962).

[10] W. Koechner, "Solid-State Lasers," Springer Verlag (1990).

[11] R F Kazarinov and R A Suris, "Amplification of electromagnetic wavesin a semiconductor superlattice," Sov. Phys. Semicond. 5, p. 707 (1971).

[12] J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, A. Y.Cho, "Quantum cascade laser," Science 264, p. 553 (1994).

325

326 BIBLIOGRAPHY

[13] B. S. Williams, H. Callebaut, S. Kumar, Q. Hu and J. L. Reno, "3.4-THzquantum cascade laser based on longitudinal-optical-phonon scatteringfor depopulation," Appl. Phys. Lett. 82, p. 1015-1017 (2003).

[14] D. Kopf, F. X. Kärtner, K. J. Weingarten, M. Kamp, and U.Keller, “Diode-pumped mode-locked Nd:glass lasers with an antireso-nant Fabry-Perot saturable absorber,” Optics Letters 20, p. 1169 (1995).

[15] J. J. Zayhowski, C. Dill, ”Diode-pumped passively Q-switched picosec-ond microchip lasers,” Opt. Lett. 19, pp. 1427 — 1429 (1994).

[16] D. J. Kuizenga and A. E. Siegman, ”FM and AM modelocking of thehomogeneous laser - part I: theory,” IEEE J. Qunat. Electron. 6, pp.694 — 701 (1970).

[17] H. A. Haus, ”A Theory of Forced Mode Locking”, IEEE Journal ofQuantum Electronics QE-11, pp. 323 - 330 (1975).

[18] H. A. Haus, ”Theory of modelocking with a fast saturable absorber,” J.Appl. Phys. 46, pp. 3049 — 3058 (1975).

[19] R. Ell, U. Morgner, F.X. Kärtner, J.G. Fujimoto, E.P. Ippen, V. Scheuer,G. Angelow, T. Tschudi: Generation of 5-fs pulses and octave-spanningspectra directly from a Ti:Sappire laser, Opt. Lett. 26, 373-375 (2001)

Chapter 7

The Dirac Formalism andHilbert Spaces

So far we treated quantum mechanics by using wave functions defined inposition space. We identified the Fourier transform of the wave function inposition space as a wave function in the wave vector or momentum space.Expectation values of operators that represent observables of the system canbe computed using either representation of the wavefunction. Obviously, thephysics must be independent whether represented in position or wave num-ber space. P.A.M. Dirac was first to introduce a representation-free notationfor the quantum mechanical state of the system and operators representingphysical observables. He realized that quantum mechanical expectation val-ues could be rewritten. For example the expected value of the Hamiltoniancan be expressed as

Zψ∗ (x) Hop ψ (x) dx = hψ|Hop |ψi , (7.1)

= hψ| ϕi , (7.2)

with

|ϕi = Hop |ψi . (7.3)

Here, |ψi and |ϕi are vectors in a Hilbert-Space, which is yet to be defined.For example, complex functions of one variable, ψ(x), that are square inte-

327

328 CHAPTER 7. THE DIRAC FORMALISM AND HILBERT SPACES

grable, i.e. Zψ∗ (x)ψ (x) dx <∞, (7.4)

form the Hilbert-Space of square integrable functions denoted as L2. In Diracnotation this is Z

ψ∗ (x)ψ (x) dx = hψ| ψi . (7.5)

Orthogonality relations can be rewritten asZψ∗m (x)ψn (x) dx = hψm| ψni = δmn. (7.6)

As the above expressions look like a bracket, he called the vector |ψni a ket-vector and hψm| a bra-vector. The meaning of these vectors clearly need aprecise mathematical definition.

7.1 Hilbert Space

A Hilbert Space is a linear vector space, i.e. if there are two elements |ϕiand |ψi in this space the sum of the elements must also be an element of thevector space

|ϕi+ |ψi = |ϕ+ ψi . (7.7)

The sum of two elements is commutative and associative

Commutative : |ϕi+ |ψi = |ψi+ |ϕi , (7.8)

Associative : |ϕi+ |ψ + χi = |ϕ+ ψi+ |χi . (7.9)

The product of the vector with a complex quantity c is again a vector of theHilbert-Space

c |ϕi ≡ |cϕi . (7.10)

The product between vectors and numbers is distributive

Distributive : c |ϕ+ ψi = c |ϕi+ c |ψi . (7.11)

In short, every linear combination of vectors in a Hilbert space is again avector in that Hilbert space.

7.1. HILBERT SPACE 329

7.1.1 Scalar Product and Norm

There is a bilinear form defined by two elments of the Hilbert Space |ϕi and|ψi , which is called a scalar product resulting in a complex number

hϕ| ψi = a . (7.12)

Ths scalar product obtained by exchanging the role of |ϕi and |ψi results inthe complex conjugate number

hψ |ϕi = hϕ |ψi ∗= a∗ . (7.13)

The scalar product is distributive

Distributive : hϕ |ψ1 + ψ2i = hϕ |ψ1i + hϕ |ψ2i . (7.14)

hϕ |cψi = c hϕ |ψi . (7.15)

And from Eq.(7.13) follows

hcψ| ϕi = hϕ| cψi∗ = c∗ hψ| ϕi . (7.16)

Thus if the complex number is pulled out from a bra-vector it becomes itscomplex conjugate. The bra- and ket-vectors are hermitian, or adjoint, toeach other. The adjoint vector is denoted by the symbol+

(|ϕi)+ = hϕ| , (7.17)

(hϕ|)+ = |ϕi . (7.18)

The vector spaces of bra- and ket-vectors are dual to each other. This meansboth sets of vectors form each of them a vector space and to each elementin one space corresponds a unique element in the other space. To transforman arbitrary expression into its adjoint, one has to replace all operators andvectors by the adjoint operator or vector and in addition the order of theelements must be reversed. For example

(c|ϕi)+ = c∗ hϕ| , (7.19)

hϕ| ψi+ = hϕ| ψi∗ = hψ| ϕi . (7.20)


This equation demands that a scalar product of a vector with itself is alwaysreal. Here, we even demand that it is positive

hϕ |ϕi > 0, real . (7.21)

The equal sign in Eq.(7.21) is only fulfilled for the null vector, which is definedby

|ϕi+ 0 = |ϕi . (7.22)

Note, we denote the null vector not with the symbol |0i but rather with thescalar 0. Because the symbol |0i is reserved for the ground state of a system.If the scalar product of a vector with itself is always positive, Eq.(7.21),

then one can derive from the scalar product the norm of a vector accordingto

kϕk =phϕ |ϕi . (7.23)

For vectors that are orthogonal to each other we have

hϕ |ψi = 0 (7.24)

without having one of them be the null vector.

7.1.2 Vector Bases

The dimensions of a Hilbert space are countable, i.e. each dimension canbe assigned a whole number and thereby all dimensions are referenced in aunique way with 1, 2, 3, ..... For example, the Hermite-Gaussian functions,which we found to be the energy eigenstates of the harmonic oscillator forma complete basis for square integrable functions on the interval [−∞,+∞].A vector space that is a Hilbert space has the following additional properties.

Completeness:

If there is a sequence of vectors in a Hilbert space |ϕ1i , |ϕ2i , |ϕ3i , |ϕ4i , ...that fulfills Cauchy’s convergence criterion then the limit vector |ϕi is alsoan element of the Hilbert space. Cauchy’s convergence criterion states thatif kϕn − ϕmk < ε, for some n,m > N(ε) the sequence converges uniformly[2].

7.2. LINEAR OPERATORS IN HILBERT SPACES 331

Separability:

The Hilbert space is separable. This indicates that for every element |ϕi inthe Hilbert space there is a sequence with |ϕi as the limit vector.Every vector in the Hilbert space can be decomposed into linear indepen-

dent basis vectors |ψni . The number of basis vectors can be infinite

|ϕi =Xn

cn |ψni . (7.25)

The components cn of the vector |ϕi with respect to the basis |ψni are com-plex numbers denoted with index n. It is advantageous to orthonormalizethe basis vectors

hψn| ψmi = δnm . (7.26)

The components of the vector |ϕi can then be determined easily by

hψm| ϕi =Xn

cn hψm| ψni =Xn

cnδmn, (7.27)

orcm = hψm| ϕi , (7.28)

This leads to|ϕi =

Xn

|ψni hψn| ϕi . (7.29)

7.2 Linear Operators in Hilbert Spaces

7.2.1 Properties of Linear Operators

An operator L is defined as a mapping of a vector |ϕi onto another vector|ψi of the Hilbert space

L |ϕi = |ψi . (7.30)

A linear operator L has the property that it maps a linear combinationof input vectors to a linear combination of the correponding individual maps

L (a |ϕ1i+ b |ϕ2i) = (a L |ϕ1i+ b L |ϕ2i)= a |ψ1i+ b |ψ2i , for a, b ∈ C. (7.31)


The sum of two linear operators is defined as

(L+M) |ϕi = L |ϕi+M |ϕi . (7.32)

And the product of two operators is defined as

L M |ϕi = L (M |ϕi) . (7.33)

The null element and 1-element of the operators is denoted as 0, and 1.Often we will not bold face these operators, especially in products, where ascalar has the same meaning. The two operators are defined by their actionon arbitrary vectors of the Hilbert space

0 |ϕi = 0, ∀ |ϕi , (7.34)

1 |ϕi = |ϕi , ∀ |ϕi (7.35)

In general, the multiplication of two operators is not commutative

L M |ϕi 6= ML |ϕi ,∀ |ϕi , (7.36)

or in shortL M 6= M L. (7.37)

The expression

[L,M] = L M − M L (7.38)

is therefore called the commutator between L and M. If [L,M] = 0, theoperators commute. The following rules for commutators apply:

[L,M] = − [M,L] , (7.39)

[L,L] = 0 , (7.40)

[L,1] = 0 , (7.41)£L,L−1

¤= 0 , (7.42)

[L,aM] = a [L,M] , (7.43)

[L1 + L2,M] = [L1,M] + [L2,M] , (7.44)

[L,M] = − [M,L] (7.45)


[L1L2,M] = [L1,M] L2 + L1 [L2,M] , (7.46)

[M,L1L2] = [M,L1] L2 + L1 [M,L2] . (7.47)

Often the anticommutator is also used. It is defined as

[L,M]+ = L M + M L . (7.48)

If [L,M]+ = 0, the operators are called anti-commuting.

7.2.2 The Dyadic Product

Two vectors in the Hilbert space can not only be used to build a scalarproduct but rather what is called a dyadic product, which is an operator

|αi hβ| . (7.49)

The dyadic product is the formal product between a ket- and a bra-vector.If applied to a vector, it projects the vector onto the state |βi and generatesa new vector in parallel to |αi with a magnitude equal to the projection

|αi hβ|ψi = hβ|ψi |αi . (7.50)

Note, the word projection has here the same meaning as in conventionalvector-calculus. There we project a vector u onto another vector v by buildingthe scaler product between the two vectors, i.e. v+u. As we have seen fromEq.(7.29), if the vectors |ψni build a complete orthonormal basis, then

|ϕi =Xn

|ψni hψn| ϕi , ∀ |ϕi , . (7.51)

Eq.(7.35) implies

1 =Xn

|ψni hψn| . (7.52)

If an operator is multiplied from the left and right side by a unit matrix, theoperator is expressed in the base |ψni

1 L 1 =Xm

|ψmi hψm|ÃX

n

L |ψni hψn|!

(7.53)

=Xm

Xn

Lmn |ψmi hψn|


with the matrix elements

Lmn = hψm|L |ψni . (7.54)

.

Ket-vector Column vector

|ϕai =Xn

an |ψni

⎛⎜⎝ a1a2...

⎞⎟⎠Bra-vector Row vector

hϕ| =Xn

a∗n hψn|¡a∗1 a∗2 · · ·

¢Inner product Scalar product

hϕa |ϕbi =Xm

Xn

a∗mbn hψm |ψni¡a∗1 a∗2 · · ·

¢·

⎛⎜⎝ b1b2...

⎞⎟⎠=Xn

a∗nbn =a∗1b1 + a∗2b2 + · · ·

Operator Matrix

L =Xm,n

Lmn |ψmi hψn|

⎛⎜⎝ L11 L12 · · ·L21 L22 · · ·...

.... . .

⎞⎟⎠Dyadic Product

|ϕai hϕb| =Xm,n

amb∗n |ψmi hψn|

⎛⎜⎝ a1a2...

⎞⎟⎠ · ¡ b∗1 b∗2 · · ·¢=

=

⎛⎜⎝ a1b∗1 a1b

∗2 · · ·

a2b∗1 a2b

∗2 · · ·

......

. . .

⎞⎟⎠The matrix elments Lmn represent the operator in the chosen base |ψni .Oncewe choose a base and represent vectors and operators in terms of this base, thecomponents of the vector and the operator can be collected in a column vectorand a matrix, respectively. The table below shows a comparison between arepresentation based on Hilbert space vectors and operators in term of vectorsand matrices in an euclidian vector space. Initially matrix mechanics was


developed by Heisenberg independently from Schroedingers wave mechanics.The Dirac representation in terms of bra- and ket-vectors unifies them andshows that both forms are isomorph to each other.

7.2.3 Special Linear Operators

7.2.4 Inverse Operators

The operator inverse to a given operator L is denoted as L−1

∀ |ϕi , |ψi = L |ϕi =⇒ |ϕi = L−1 |ψi , (7.55)

which leads to

LL−1 = 1 (7.56)

The inverse of a product is the product of the inverse in inverse order

(ML)−1 = L−1M−1 (7.57)

7.2.5 Adjoint or Hermitian Conjugate Operators

The adjoint or hermitian conjugate operator L+ is defined by

hψ|L |ϕi∗ =L+ψ

¯ϕi∗ = hϕ|L+ |ψi , (7.58)

here |ϕi and |ψi are arbitrary vectors in the Hilbert space. Note, that if theadjoint of an expression is formed, each component gets conjugated and theorder is reversed. For example

(L |ϕi)+ = hϕ|L+, (7.59)¡L+ |ϕi

¢+= hϕ|L (7.60)

Given a certain base |ψni , the matrix elements of the adjoint operator inthat basis follow directly from the definition (7.58)

L+mn = hψm|L+ |ψni = hψn|L |ψmi∗ = L∗nm (7.61)

The following rules apply to adjoint operators¡L+¢+= L, (7.62)


(aL)+ = a∗L+, (7.63)

(L+M)+ = L++M+, (7.64)

(L M)+ =M+ L+. (7.65)

7.2.6 Hermitian Operators

If the adjoint operator L+ is identical to the operator itself, then we call theoperator hermitian

L = L+. (7.66)

Hermitian operators have real expected values. As discussed already duringour exploration of wave mechanics, physical observables are represented byhermitian operators.

7.2.7 Unitary Operators

If the inverse of an operator U is the adjoint operator

U−1= U+, (7.67)

then this operator is called a unitary operator and

U+U = UU+= 1. (7.68)

If the operator U is unitary and H is a hermitian operator, then the productUHU−1 is also a hermitian operator.¡

UHU−1¢+=¡U−1

¢+H+U+ = UHU−1. (7.69)

7.2.8 Projection Operators

The dyadic productPn = |ψni hψn| , (7.70)

is a projection operator Pn that projects a given state |ϕi onto the unitvector |ψni

Pn |ϕi = |ψni hψn|ϕi . (7.71)

7.3. EIGENVALUES OF OPERATORS 337

If |ϕi is represented in the orthonormal base |ψni

|ϕi =Xn

cn |ψni , (7.72)

we obtainPn |ϕi = cn |ψni . (7.73)

By construction, projection operators are hermitian operators. Besides op-erators that project on vectors, there are also operators that project on sub-spaces of the Hilbert space

PU =XU

|ψni hψn| . (7.74)

Here, the orthonormal vectors |ψni span the sub-space U . Projection opera-tors are idempotent

Pkn = Pn, for k > 1. (7.75)

7.3 Eigenvalues of Operators

In chapter 4, we studied the eigenvalue problem of differential operators.Here, we want to formulate the eigenvalue problem of operators in a Hilbertspace. An operator L in a Hilbert space with eigenvectors |ψni fulfills theequations

L |ψni = Ln |ψni , (7.76)

with eigenvalues Ln. If there exist several different eigenvectors to the sameeigenvalue Ln, this eigenvalue is called degenerate. For example, the energyeigenfunctions of the hydrogen atom are degenerate with respect to the in-dices l and m. The set of all eigenvalues is called the eigenvalue spectrumof the operator L. As shown earlier the eigenvalues of hermitian operatorsare real and the eigenvectors to different eigenvalues are orthogonal to eachother, because

hψm|L |ψni = Ln hψm |ψni = Lm hψm |ψni , (7.77)

or(Ln − Lm) hψm |ψni = 0. (7.78)


If the eigenvectors of the operator L form a complete base of the Hilbertspace, the operator L is represented in this base by a diagonal matrix

Lmn = hψm|L |ψni = Ln hψm |ψni = Ln δmn (7.79)

The operator can then be written in its spectral representation

L =Xn

Ln |ψni hψn| =Xn

LnPn . (7.80)

7.4 Eigenvectors of Commuting Operators

Two operators, A and B, that commute with each other have a common setof eigenvectors. To proove this theorem, we assume that the eigenvalue spec-trum of the operator A is non degenerate. The eigenvectors and eigenvaluesof operator A are |ψni and An, respectively

A |ψni = An |ψni . (7.81)

Using[A,B] = 0, (7.82)

we find

hψm|AB−BA |ψni = 0 (7.83)

hψm|AÃX

n

|ψni hψn|!B−BA |ψni = 0

(Am −An) hψm|B |ψni = (Am −An)Bmn = 0 .

Since the eigenvalues are assumed to be not degenerate, i.e. Am 6= An, thematrix Bmn must be diagonal, which means that the vector |ψni has also tobe an eigenvector of operator B. If the eigenvalues are degenerate, one canalways choose, in the sub-space that belongs to the degenerate eigenvalue, abase that are also eigenvectors of B. The operator B thus eventually has nodegeneracies in this sub-space and therefore, the eigenvalues of B may helpto uniquely characterize the set of joint eigenvectors.Also the reverse is true. If two operators have a joint system of eigenvec-

tors, they commute. This is easy to see from the spectral representation ofboth operators.

7.5. COMPLETE SYSTEM OF COMMUTING OPERATORS 339

Example: We define the parity operator Px which, when applied to awave function of a particle in one dimension ψ(x), changes the sign of theposition x

Pxψ(x) = ψ(−x). (7.84)

The Hamiltonian of a particle in an inversion symmetric potential V (x), i.e.

V (x) = V (−x), (7.85)

commutes with the parity operator. Then the eigenfunctions of the Hamilto-nian are also eigenfunctions of the parity operator. The eigenfunctions of theparity operator are the symmetric or antisymmetric functions with eigenval-ues 1 and −1, respectively. Therefore, the eigenfunctions of a Hamiltonianwith symmetric potential has symmetric and antisymmetric eigenfunctions,see box potential and harmonic oscillator potential.

7.5 Complete System of Commuting Opera-tors

In the case of the hydrogen atom, we had to use three qunatum numbersn, l, and m to characterize the energy eigenfunctions completely. Withoutproof, the indices l and m characterize the eigenvalues of the square of theangular momentum operator L2, and of the z-component of the angular mo-ment Lz with eigenvalues l (l + 1) ~2 and m~, respectively. One can show,that the Hamilton operator of the hydrogen atom, the square of the angularmomentum operator and the z-component of the angular moment operatorcommute with each other and build a complete system of commuting op-erators (CSCO), whose eigenvalues enable a unique characterization of theenergy eigenstates of the hydrogen atom.

7.6 Product Space

Very often in quantum mechanics one deals with interacting systems, forexample system A and system B. The state space of each isolated systemis Hilbert space HA and Hilbert space HB spanned by a complete base |ψniA and |ψni B, respectively. LA and MB are operators on each of the Hilber


spaces of the individual systems. The Hilbert space of the total system isthe product space

H = HA ⊗HB. (7.86)

The vectors in this Hilbert space are given by the direct product of theindividual vectors and one could choose as a base in the product space

|χnmi = |ψniA ⊗ |ψmiB = |ψniA |ψmiB . (7.87)

Operators that only act on system A can be extended to operate on theproduct space by

L = LA ⊗ 1B. (7.88)

or similar for operators acting on system B

M = 1A ⊗MB. (7.89)

The product of both operators is then

LM = LA ⊗MB. (7.90)

An operator in this product space acts on a vector in the following way

LM |χnmi = L |ψniA ⊗M |ψmiB . (7.91)

Since, the vecotrs |ψniA and |ψmiB build a complete base for system A andB, respectively, the product vectors in Eq.(7.87) build a complete base forthe interacting system and each state can be written in terms of this base

|χi =Xm,n

amn |χnmi =Xm,n

amn |ψniA ⊗ |ψmiB . (7.92)

7.7 Quantum Dynamics

In chapter 4, we derived the Schroedinger Equation in the x-represenation.The stationary Schroedinger Equation was written as an eigenvalue problemto the Hamiltonian operator, which was then a differential operator. Withthe Dirac formulation we can rewrite these equations without refering to aspecial representation.

7.7. QUANTUM DYNAMICS 341

7.7.1 Schroedinger Equation

In the Dirac notation the Schroedinger Equation is

j~∂ |Ψ (t)i

∂t= H |Ψ (t)i . (7.93)

H is the Hamiltonian operator; it determines the dynamics of the quantumsystem.

H =p2

2m+V(x). (7.94)

The Hamiltonian operator is the generator of motion in a quantum system.Here p and x and functions of them are operators in the Hilbert space. |Ψ (t)iis the Hilbert space vector describing fully the system’s quantum state at timet. When looking for states that have a harmonic temporal behaviour

|Ψ (t)i = e−jEnt/~ |ψni , (7.95)

we obtain the stationary Schroedinger Equation

H |ψni = En |ψni , (7.96)

that determines the energy eigenstates of the system. If the |ψni build acomplete basis of the Hilbert space, H, the system is dynamically evolving,the most general time dependent solution of the Schroedinger Equation isthen a superposition of all energy eigenstates

|Ψ (t)i =Xn

an e−jEnt/~ |ψni . (7.97)

7.7.2 Schroedinger Equation in x-representation

We can return to wave mechanics by rewriting the abstract SchroedingerEquation in the eigenbase |xi of the position operator. For simplicity innotation, we only consider the one dimensional case and define that thereexists the following eigenvectors

x |x0i = x0 |x0i , (7.98)

with the orthogonality relation

hx |x0i = δ(x− x0). (7.99)


Note, since the position operator has a continuous spectrum of eigenvaluesthe orthogonality relation is a dirac delta function rather than a delta-symbol.The completness relation using this base is expressed in the unity operator

1 =

Z|x0i hx0| dx0 , (7.100)

rather then a sum as in Eq.(7.52). Inserting this unity operator in theSchroedinger Equation (7.93) and projecting from the left with hx|, we obtain

j ~∂

∂thx |Ψ (t)i = hx| H

Z|x0i hx0|Ψ (t) i dx0. (7.101)

The expression hx |Ψ (t)i is the probability amplitude that a position mea-surement on the system in state |Ψ (t)i yields the value x, which is preciselythe meaning of the wave function

Ψ (x, t) = hx |Ψ (t)i (7.102)

in chapter 4. Using the eigenvalue property of the states and the orthogo-nality relations we obtain from Eq.(7.101)

j ~ ∂∂tΨ (x, t) = H(x, p =

~j∂

∂x)Ψ (x, t) . (7.103)

7.7.3 Canonical Quantization

Thus the dynamics of a quantum system is fully determined by its Hamil-tonian operator. The Hamiltonian operator is usually derived from theclassical Hamilton function according to the Hamilton-Jacobi formulation ofClassical Mechanics [3]. The classical Hamilton function H(qi , pi) is afunction of the position coordinates of a particle xi or generalized coordinatesqi and the corresponding momentum coordinates pi. The classical equationsof motion are found by

qi(t) =∂

∂piH(qi , pi) , (7.104)

pi(t) = − ∂

∂qiH(qi , pi) . (7.105)

In quantum mechanics the Hamiltonian function and the position andmomentum coordinates become operators and quantization is achieved by

7.7. QUANTUM DYNAMICS 343

imposing on position and momentum operators that are related to the samedegree of freedom, for example the x-coordinate of a particle and the associatemomentum px, canonical commutation relations

H(qi , pi) ⇒ H(qi , pi), (7.106)

[qi,pj] = j ~δij. (7.107)

Imposing this commutation relation implies that position and momentumrelated to one degree of freedom can not be measured simultaneously witharbitrary precision and Heisenberg’s uncertainty relation applies to the pos-sible states the system can take on.

7.7.4 Schroedinger Picture

In the Schroedinger picture the quantum mechanical state of the system isevolving with time. If there is no explicit time dependence in the operatorsthen the operators stay time independent. The Schroedinger Equation (7.93)

j~∂ |Ψ (t)i

∂t= H |Ψ (t)i , (7.108)

plus the initial state

|Ψ (t = 0)i = |Ψ (0)i , (7.109)

unquely determine the dynamics of the system. The evolution of the quantumstate vector can be described as a mapping of the initial state by a timeevolution operator.

|Ψ (t)i = U(t) |Ψ (0)i . (7.110)

If this solution is substituted into the Schroedinger Equation (7.108) it followsthat the time evolution operator has to obey the equation

j ~∂

∂tU(t) = H U(t). (7.111)

For a time independent Hamiltonian Operator the formal integration of thisequation is

U(t) = exp [−jHt/~] . (7.112)


The time evolution operator is unitary

U−1(t) = U+(t) , (7.113)

because the Harmiltonian operator is hermitian, and therefore the norm ofan initial state is preserved. The initial value for the time evolution operatoris

U(t = 0) = 1. (7.114)

The expected value of an arbitrary operator A is given by

hΨ (t)| A |Ψ (t)i = hΨ (0)|U+(t)AU(t) |Ψ (0)i . (7.115)

7.7.5 Heisenberg Picture

Since the physically important quantities are the expected values, i.e. theoutcome of experiments, Eq.(7.115) can be used to come up with an alter-native formulation of quantum mechanics. In this formulation, called theHeisenberg picture, the operators are evolving in time according to

AH(t) = U+(t)AU(t), (7.116)

and the state of the system is time independent and equal to its initial state

|ΨH (t)i = |ΨS (0)i . (7.117)

Clearly an expected value for a time dependent operator using the Heisenbergstate (7.117) is identical with Eq.(7.113).This is identical to describing a unitary process in an eucledian vector

space. Scalar products between vectors are preserved, if all vectors are un-dergoing a unitary transformation, i.e. a rotation for example. An alternativedescription is that the vectors are time independent but the coordinate sys-tem rotates in the opposite direction. When the coordinate system changes,the operators described in the time dependent coordinate system becometimedependent themselves.From the definition of the time evolution operator we find immediately

an equation of motion for the time dependent operators of the Heisenbergpicture

j~∂AH(t)

∂t=

µj~

∂U+(t)

∂t

¶ASU(t) +U

+(t)AS

µj ~

∂U(t)

∂t

¶(7.118)

+U+(t)

µj~∂AS

∂t

¶U(t)

7.8. THE HARMONIC OSCILLATOR 345

j~∂AH(t)

∂t= −U+(t)H+ASU(t) +U

+(t)ASHU(t) (7.119)

+U+(t)

µj~∂AS

∂t

¶U(t)

Using the relation U+(t)U(t) = U(t)U+(t) = 1 and inserting it betweenthe Hamiltonian operator and the operator A, we finally end up with theHeisenberg equations of motion for the Heisenberg operators

j~∂

∂tAH(t) = −HHAH +AHHH + j~

µ∂A

∂t

¶H

(7.120)

= [AH ,HH ] + j ~µ∂A

∂t

¶H

(7.121)

with

HH = U+(t)HSU(t) (7.122)

= HS for conservative systems, i.e. HS 6= HS(t) (7.123)

Note, that the last term in Eq.(7.121) is only present if the Schroedingeroperators do have an explicit time dependence, a case which is beyond thescope of this class.

7.8 The Harmonic Oscillator

To illustrate the beauty and efficiency in describing the dynamics of a quan-tum system using the dirac notation and operator algebra, we reconsider theone-dimensional harmonic oscillator discussed in section 4.4.2 and describedby the Hamiltonian operator

H =p2

2m+1

2K x2, (7.124)

with[x,p] = j~. (7.125)


7.8.1 Energy Eigenstates, Creation and AnnihilationOperators

It is advantageous to introduce the following normalized position and mo-mentum operators

X =

rK

~ω0x (7.126)

P =1√

m~ω0p (7.127)

with ω0 =q

Km. The Hamiltonian operator and the commutation relationship

of the normalized position and momentum operator resume the simpler forms

H =~ω02

¡P2 +X2

¢, (7.128)

[X,P] = j . (7.129)

Algebraically, it is very useful to introduce the nonhermitian operators

a =1√2(X+ jP) , (7.130)

a+ =1√2(X−jP) , (7.131)

which satisfy the commutation relation£a,a+

¤= 1. (7.132)

We find

aa+ =1

2

¡X2 +P2

¢− j2[X,P] =

1

2

¡X2 +P2 + 1

¢, (7.133)

a+a =1

2

¡X2 +P2

¢+j

2[X,P] =

1

2

¡X2 +P2 − 1

¢, (7.134)

and the Hamiltonian operator can be rewritten in terms of the new operatorsa and a+ as


H =~ω02

¡a+a+ aa+

¢(7.135)

= ~ω0µa+a+

1

2

¶. (7.136)

We introduce the operatorN = a+a, (7.137)

which is a hermitian operator. Up to an additive constant 1/2 and a scalingfactor ~ω0 equal to the energy of one quantum of the harmonic oscillator itis equal to the Hamiltonian operator of the harmonic oscillator. Obviously,N is the number operator counting the number of energy quanta excited in aharmonic oscillator. We assume that the number operatorN has eigenvectorsdenoted by |ni and corresponding eigenvalues n

N |ni = a+a |ni = n |ni . (7.138)

We also assume that these eigenvectors are normalized and since N is her-mitian they are also orthogonal to each other

hm |ni = δmn. (7.139)

Multiplication of this equation with the operator a and use of the commuta-tion relation (7.132) leads to

a a+a |ni = na |ni (7.140)¡a+a+ 1

¢a |ni = na |ni (7.141)

N a |ni = (n− 1)a |ni (7.142)

Eq.(7.140) indicates that if |ni is an eigenstate to the number operator Nthen the state a |ni is a new eigenstate to N with eigenvalue n− 1. Becauseof this property, the operator a is called a lowering operator or annihilationoperator, since application of the annihilation operator to an eigenstate withn quanta leads to a new eigenstate that contains one less quantum

a |ni = C |n− 1i , (7.143)

where C is a yet undetermined constant. This constant follows from thenormalization of this state and being an eigenvector to the number operator.

hn| a+a |ni = |C|2 , (7.144)

C =√n. (7.145)


Thus

a |ni =√n |n− 1i . (7.146)

Clearly, if there is a state with n = 0 application of the annihilation operatorleads to the null-vector in this Hilbert space, i.e.

a |0i = 0, (7.147)

and there is no other state with a lower number of quanta. Thus, the n’smust be whole numbers and |0i is the ground state of the harmonic oscillator,the state with the lowest energy, otherwise the iteration (7.146) would notstop.If a is an annihilation operator for energy quanta, a+ must be a cre-

ation operator for energy quanta, otherwise the state |ni would not fulfillthe eigenvalue equation Eq.(7.138)

a+a |ni = n |ni (7.148)

a+√n |n− 1i = n |ni (7.149)

a+ |n− 1i =√n |ni (7.150)

or

a+ |ni =√n+ 1 |n+ 1i . (7.151)

Starting from the energy ground state of the harmonic oscillator |0i withenergy ~ω0/2 we can generate the n-th energy eigenstate by n-fold applicationof the creation operator a+ and proper normalization

|ni = 1p(n+ 1)!

¡a+¢n |0i , (7.152)

with

H |ni = En |ni , (7.153)

and

En = ~ω0µn+1

2

¶. (7.154)


7.8.2 Matrix Representation

We can express the normalized position and momentum operators as func-tions of the creation and annihilation operators

X =1√2

¡a+ + a

¢, (7.155)

P =j√2

¡a+−a

¢. (7.156)

These operators do have the following matrix representations

hm| a |ni =√nδm,n−1 , hm| a+ |ni =

√n+ 1δm,n+1 , (7.157)

hm| a+a |ni = nδm,n , hm| aa+ |ni = (n+ 1) δm,n , (7.158)

hm|X |ni =1√2

³√n+ 1δm,n+1 +

√nδm,n−1

´, (7.159)

hm|P |ni =j√2

³√n+ 1δm,n+1 −

√nδm,n−1

´, (7.160)

hm| a2 |ni =pn(n− 1)δm,n−2 , (7.161)

hm| a+2 |ni =p(n+ 1) (n+ 2)δm,n+2 , (7.162)

hm|X2 |ni =1

2

µ(2n+ 1)δm,n +

pn(n− 1)δm,n−2

+p(n+ 1) (n+ 2)δm,n+2

¶, (7.163)

hm|P2 |ni =1

2

µ(2n+ 1)δm,n −

pn(n− 1)δm,n−2

−p(n+ 1) (n+ 2)δm,n+2

¶. (7.164)

7.8.3 Minimum Uncertainty States or Coherent States

From the matrix elements calculated in the last section, we find that theenergy or quantum number eigenstates |ni have vanishing expected valuesfor position and momentum. This also follows from the x-representationψn(x) = hx |ni studied in section 4.4.2

hn|X |ni = 0 , hn|P |ni = 0 , (7.165)

and the fluctuations in position and momentum are then simply

hn|X2 |ni = n+1

2, hn|P2 |ni = n+

1

2. (7.166)


The minimum uncertainty product for the fluctuations

∆X =

qhn|X2 |ni− hn|X |ni2 = n+

1

2, (7.167)

∆P =

qhn|P2 |ni− hn|P |ni2 = n+

1

2. (7.168)

is then∆X ·∆P = n+

1

2. (7.169)

Only the ground state n = 0 is a minimum uncertainty wave packet, since itsatisfies the eigenvalue equation

a |0i = 0, (7.170)

wherea =

1√2(X+ jP) , . (7.171)

In fact we can show that every eigenstate to the annihilation operator

a |αi = α |αi , for α C (7.172)

is a minimum uncertainty state. We obtain for expected values of positionor momentum in these states

hα|a |αi = α , hα| a+ |αi = α∗, (7.173)

hα|X |αi =1√2(α + α∗) , hα|P |αi = j√

2(α − α∗) , (7.174)

and for its squares

hα|a+a |αi = |α|2 , hα| aa+ |αi =¡|α|2 + 1

¢, (7.175)

hα| a2 |αi = α2, hα| a+2 |αi = α∗2 , (7.176)

hα|X2 |αi =1

2

¡α2 + 2α∗α+ α∗2 + 1

¢= hα|X |αi2 + 1

2, (7.177)

hα|P2 |αi =1

2

¡−α2 + 2α∗α− α∗2 + 1

¢= hα|P |αi 2 + 1

2. (7.178)

Thus the uncertainty product is at its minimum


∆X ·∆P =1

2∀ α C. (7.179)

In fact one can show that the statistics of a position or momentum measure-ment for a harmonic oscillator in this state follows a Gaussian satistics withthe average and variance given by Eqs.(7.173), (7.177) and (7.178). This canbe represented pictorially in a phase space diagram as shown in Figure 7.1

<P>

<X> X

2ΔP

2ΔXP

0

α

Figure 7.1: Representation of a minimum uncertainty state of the harmonicoscillator as a phase space distribution.

7.8.4 Heisenberg Picture

The Heisenberg equations of motion for a linear system like the harmonicoscillator are linear differential equations for the operators, which can beeasily solved. From Eqs.(7.121) we find

j~ ∂∂taH(t) = [aH ,H] (7.180)

= ~ω0aH , (7.181)

with the solutionaH(t) = e−jω0taS . (7.182)

Therefore, the expectation values for the creation, annihilation, position andmomentum operators are identical to those of Eqs.(7.173) to (7.178); we only


need to subsitute α→ αe−jω0t . We may again pictorially represent the timeevolution of these states as a probability distribution in phase space, seeFigure 7.2.

<P>

<X> X

2ΔP

2ΔX

P

0

α

αe-j tω O

Figure 7.2: Time evolution of a coherent state in phase space.

7.9 The Kopenhagen Interpretation of Quan-tum Mechanics

7.9.1 Description of the State of a System

At a given time t the state of a system is described by a normalized vector|Ψ(t)i in the Hilbert space, H. The Hilbert space is a linear vector space.Therefore, any linear combination of vectors is again a possible state of thesystem. Thus superpositions of states are possible and with it come interfer-ences.

7.9.2 Description of Physical Quantities

Measurable physical quantities, observables, are described by hermitian op-erators A = A+.

7.9. THEKOPENHAGEN INTERPRETATIONOFQUANTUMMECHANICS353

7.9.3 The Measurement of Observables

An observable has a spectral representation in terms of eigenvectors andeigenvalues, which can be discrete or continuous, here we discuss the discretecase

A =Xn

An |Ani hAn| , (7.183)

The eigenvectors are orthogonal to each other and the eigenvalues are real

hAn| An0i = δn,n0 . (7.184)

Upon a measurement of the observable A of the system in state |Ψ(t)i theoutcome can only be one of the eigenvalues An of the observable and theprobability for that event to occur is

pn = | hAn| Ψ(t)i|2 . (7.185)

If the eigenvalue spectrum of the operator A is degenerate, the probabilitiesof the probabilities of the different states to the same eigenvector need to beadded.After the measurement the system is in the eigenstate |Ani corresponding

to the eigenvalue An found in the measurement, which is called the reduc-tion of state[1]. This unphysical reduction of state is only necessary as ashortcut for the description of the measurement process and the fact thatthe system becomes entangled with the state of the macroscopic measure-ment equipment. This entanglement leads to a necessary decoherence of thesuperposition state of the measured system, which is equivalent to assuminga reduced state.

Bibliography

[1] Introduction to Quantum Mechanics, Griffiths, David J., Prentice Hall,1995.

[2] Functional Analysis, G. Bachman and L. Naricci, Academic Press, 1966.

[3] Classical Mechanics, H. Goldstein, Addison and Wesley series in physics,1959.


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